Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 857070 11 pageshttpdxdoiorg1011552013857070

Research ArticleCertain Types of Interval-Valued Fuzzy Graphs

Muhammad Akram1 Noura Omair Alshehri2 and Wieslaw A Dudek3

1 Punjab University College of Information Technology University of the Punjab Old Campus Lahore 54000 Pakistan2Department of Mathematics Faculty of Sciences (Girls) King Abdulaziz University Jeddah Saudi Arabia3 Institute of Mathematics and Computer Science Wroclaw University of Technology Wybrzeze Wyspianskiego 2750-370 Wroclaw Poland

Correspondence should be addressed to Noura Omair Alshehri nalshehriekauedusa

Received 29 July 2013 Accepted 31 August 2013

Academic Editor Chong Lin

Copyright copy 2013 Muhammad Akram et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We propose certain types of interval-valued fuzzy graphs including balanced interval-valued fuzzy graphs neighbourly irregularinterval-valued fuzzy graphs neighbourly total irregular interval-valued fuzzy graphs highly irregular interval-valued fuzzy graphsand highly total irregular interval-valued fuzzy graphs Some interesting properties associated with these new interval-valued fuzzygraphs are investigated and necessary and sufficient conditions under which neighbourly irregular and highly irregular interval-valued fuzzy graphs are equivalent are obtained We also describe the relationship between intuitionistic fuzzy graphs and interval-valued fuzzy graphs

1 Introduction

Themajor role of graph theory in computer applications is thedevelopment of graph algorithms A number of algorithmsare used to solve problems that are modeled in the formof graphs These algorithms are used to solve the graphtheoretical concepts which in turn are used to solve the cor-responding computer science application problems Severalcomputer programming languages support the graph theoryconcepts [1] The main goal of such languages is to enablethe user to formulate operations on graphs in a compact andnatural manner Some of these languages are (1) SPANTREEto find a spanning tree in the given graph (2) GTPL graphtheoretic language (3) GASP graph algorithm softwarepackage (4) HINT an extension of LISP (5) GRASPE anextension of LISP (6) IGTS an extension of FORTRAN (7)GEA graphic extended AL-GOL (8) AMBIT to manipulatedigraphs (9) GIRL graph information retrieval languageand (10) FGRAAL FORTRAN Extended graph algorithmiclanguage [1 2]

Zadeh [3] introduced the notion of interval-valued fuzzysets and Atanassov [4] introduced the concept of intuition-istic fuzzy sets as extensions of Zadehrsquos fuzzy set theory [5]

for representing vagueness and uncertainty Interval-valuedfuzzy set theory reflects the uncertainty by the length of theinterval membership degree [120583

1 1205832] In intuitionistic fuzzy

set theory for every membership degree (1205831 1205832) the value

120587 = 1 minus 1205831

minus 1205832denotes a measure of nondeterminacy

(or undecidedness) Interval-valued fuzzy sets provide amore adequate description of vagueness than traditionalfuzzy sets It is therefore important to use interval-valuedfuzzy sets in applications such as fuzzy control One ofthe computationally most intensive parts of fuzzy control isdefuzzification [6] Since interval-valued fuzzy sets are widelystudied and used we describe briefly thework of Gorzalczanyon approximate reasoning [7 8] Roy and Biswas on medicaldiagnosis [9] Turksen onmultivalued logic [10] andMendelon intelligent control [6]

Kauffmanrsquos initial definition of a fuzzy graph [11] wasbased on Zadehrsquos fuzzy relations [5] Rosenfeld [12] intro-duced the fuzzy analogue of several basic graph-theoreticconcepts Since then fuzzy graph theory has been findingan increasing number of applications in modeling real timesystems where the level of information inherent in the systemvaries with differences levels of precision Fuzzy modelsare becoming useful because of their aim to reduce the

2 Journal of Applied Mathematics

differences between the traditional numerical models used inengineering and sciences and the symbolic models used inexpert systems Mordeson and Peng [13] defined the conceptof complement of fuzzy graph and described some operationson fuzzy graphs In [14] the definition of complement ofa fuzzy graph was modified so that the complement of thecomplement is the original fuzzy graph which agreeswith thecrisp graph case Ju andWang gave the definition of interval-valued fuzzy graph in [15] Akram et al [16ndash20] introducedmany new concepts including bipolar fuzzy graphs interval-valued line fuzzy graphs and strong intuitionistic fuzzygraphs In this paper we propose certain types of interval-valued fuzzy graphs including balanced interval-valued fuzzygraphs neighbourly irregular interval-valued fuzzy graphsneighbourly total irregular interval-valued fuzzy graphshighly irregular interval-valued fuzzy graphs and highlytotal irregular interval-valued fuzzy graphs Some interestingproperties associated with these new interval-valued fuzzygraphs are investigated and necessary and sufficient condi-tions under which neighbourly irregular and highly irregularinterval-valued fuzzy graphs are equivalent are obtained Wealso describe the relationship between intuitionistic fuzzygraphs and interval-valued fuzzy graphs

We used standard definitions and terminologies in thispaper For notations terminologies and applications are notmentioned in the paper the readers are referred to [13 14 21ndash29]

2 Preliminaries

In this section we review some elementary concepts whoseunderstanding is necessary to fully benefit from this paper

By a graph 119866lowast = (119881 119864) we mean a nontrivial finite

connected and undirected graph without loops or multipleedges We write 119909119910 isin 119864 to mean (119909 119910) isin 119864 and if 119890 = 119909119910 isin 119864we say that 119909 and 119910 are adjacent Formally given a graph119866lowast = (119881 119864) two vertices 119909 119910 isin 119881 are said to be neighbors

or adjacent nodes if 119909119910 isin 119864 The number of vertices thecardinality of 119881 is called the order of graph and denotedby |119881| The number of edges the cardinality of 119864 is calledthe size of graph and denoted by |119864| A path in a graph 119866

lowast

is an alternating sequence of vertices and edges V0 1198901 V1

1198902 V

119899minus1 119890119899 V119899 The path graph with 119899 vertices is denoted

by 119875119899 A path is sometimes denoted by 119875

119899 V0V1sdot sdot sdot V119899(119899 gt 0)

The length of a path 119875119899in119866lowast is 119899 A path 119875

119899 V0V1sdot sdot sdot V119899in119866lowast

is called a cycle if V0= V119899and 119899 ge 3 Note that path graph 119875

119899

has 119899 minus 1 edges and can be obtained from a cycle graph 119862119899

by removing any edge An undirected graph 119866lowast is connected

if there is a path between each pair of distinct vertices Theneighbourhood of a vertex V in a graph 119866

lowast is the inducedsubgraph of 119866lowast consisting of all vertices adjacent to V andall edges connecting two such vertices The neighbourhoodis often denoted 119873(V) The degree deg(V) of vertex V is thenumber of edges incident on V or equivalently deg(V) =

|119873(V)| The set of neighbors called a (open) neighborhood119873(V) for a vertex V in a graph 119866

lowast consists of all verticesadjacent to V but not including V that is119873(V) = 119906 isin 119881 | V119906 isin

119864 When V is also included it is called a closed neighborhood119873[V] that is 119873[V] = 119873(V) cup V A regular graph is a graph

where each vertex has the same number of neighbors that isall the vertices have the same closed neighbourhood degreeA connected graph is highly irregular if each of its vertices isadjacent only to vertices with distinct degrees Equivalentlya graph 119866

lowast is highly irregular if every two vertices of 119866lowast

connected by a path of length 2 have distinct degrees Aconnected graph is said to be neighbourly irregular if no twoadjacent vertices of 119866lowast have the same degree Equivalently aconnected graph 119866lowast is called neighbourly irregular if everytwo adjacent vertices of 119866lowast have distinct degree

It is known that one of the best known classes of graphs isthe class of regular graphs These graphs have been studiedextensively in various contexts Regular graphs of degree 119903

and order 119899 exist with only limited but natural restrictionsIndeed for integers 119903 and 119899 with 0 le 119903 le 119899 minus 1 an 119903-regulargraph of order 119899 exists if and only if 119899119903 is even A graph thatis not regular will be called irregular It is well known [30]that all nontrivial graphs regular or irregular must containat least two vertices of the same degree In a regular graphof course every vertex is adjacent only to vertices having thesame degree On the other hand it is possible for a vertex inan irregular graph to be adjacent only to vertices with distinctdegrees With these observations made we now considergraphs that are opposite in a certain sense to regular graphsWe consider only undirected graphs with the finite numberof vertices and edges

Applications of fuzzy relations arewidespread and impor-tant especially in the field of clustering analysis neuralnetworks computer networks pattern recognition decisionmaking and expert systems In each of these the basicmathematical structure is that of a fuzzy graph

Definition 1 (see [3 5]) A fuzzy subset 120583 on a set 119883 is a map120583 119883 rarr [0 1] A fuzzy binary relation on119883 is a fuzzy subset120583 on 119883 times 119883 By a fuzzy relation we mean a fuzzy binaryrelation given by 120583 119883 times 119883 rarr [0 1]

Fuzzy set theory is an extension of ordinary set theoryin which to each element a real number between 0 and 1called the membership degree is assigned Unfortunately itis not always possible to give an exact degree of membershipThere can be uncertainty about the membership degreebecause of lack of knowledge vague information and soforth A possible way to overcome this problem is to useinterval-valued fuzzy sets which assign to each element aclosed interval which approximates the ldquorealrdquo but unknownmembership degree The length of this interval is a measurefor the uncertainty about the membership degree

An interval number119863 is an interval [119886minus 119886+]with 0 le 119886minus le

119886+ le 1 The interval [119886 119886] is identified with the number 119886 isin

[0 1] Let119863[0 1] be the set of all closed subintervals of [0 1]Then it is known that (119863[0 1] le or and) is a complete latticewith [0 0] as the least element and [1 1] as the greatest

Definition 2 (see [7]) An interval-valued fuzzy relation 119877 ina universe119883times119884 is a mapping 119877 119883times119884 rarr 119863[0 1] such that119877(119909 119910) = [119877minus(119909 119910) 119877+(119909 119910)] isin 119863[0 1] for all pairs (119909 119910) isin

119883 times 119884

Journal of Applied Mathematics 3

Interval-valued fuzzy relations reflect the idea that mem-bership grades are often not precise and the intervals repre-sent such uncertainty

Definition 3 (see [15]) By an interval-valued fuzzy graph119866 ofa graph 119866

lowast we mean a pair 119866 = (119860 119861) where 119860 = [120583minus119860 120583+119860]

is an interval-valued fuzzy set on 119881 and 119861 = [120583minus119861 120583+119861] is an

interval-valued fuzzy relation on 119864 such that

120583minus

119861(119909119910) le min (120583

minus

119860(119909) 120583

minus

119860(119910))

120583+

119861(119909119910) le min (120583

+

119860(119909) 120583

+

119860(119910))

(1)

for all 119909119910 isin 119864

Throughout this paper 119866lowast is a crisp graph and 119866 is aninterval-valued fuzzy graph

3 Balanced Interval-Valued Fuzzy Graphs

Definition 4 Let 119866 be an interval-valued fuzzy graph Theneighbourhood degree of a vertex 119909 in119866 is defined by deg(119909) =

[deg120583minus

(119909) deg120583+

(119909)] where deg120583minus

(119909) = sum119910isin119873(119909)

120583minus119860(119910) and

deg120583+

(119909) = sum119910isin119873(119909)

120583+119860(119910) Notice that 120583minus

119861(119909119910) gt 0 120583+

119861(119909119910) gt

0 for 119909119910 isin 119864 and 120583minus119861(119909119910) = 120583+

119861(119909119910) = 0 for 119909119910 notin 119864

Definition 5 Let 119866 = (119860 119861) be an interval-valued fuzzygraph on119866lowast If all the vertices have the same open neighbour-hood degree 119899 then 119866 is called an 119899-regular interval-valuedfuzzy graph The open neighbourhood degree of a vertex 119909

in 119866 is defined by deg(119909) = [deg120583minus

(119909) deg120583+

(119909)] wheredeg120583minus

(119909) = sum119910isin119873(119909)

120583minus119860(119910) and deg

120583+

(119909) = sum119910isin119873(119909)

120583+119860(119910)

Definition 6 Let 119866 be an interval-valued fuzzy graph Theclosed neighbourhood degree of a vertex 119909 is defined bydeg[119909] = [deg

120583minus

[119909] deg120583+

[119909]] where

deg120583minus[119909] = deg

120583minus(119909) + 120583

minus

119860(119909)

deg120583+[119909] = deg

120583+(119909) + 120583

+

119860(119909)

(2)

If all the vertices have the same closed neighbourhood degree119898 then 119866 is called a 119898-totally regular interval-valued fuzzygraph

Example 7 Consider a graph 119866lowast such that 119881 = 119886 119887 119888 119889119864 = 119886119887 119887119888 119888119889 119886119889 Let 119860 be an interval-valued fuzzy subsetof 119881 and let119861 be an interval-valued fuzzy subset of119864 sube 119881times119881

defined by

[[[

[

119886 119887 119888 119889

120583minus119860

03 03 03 03

120583+119860

05 05 05 05

]]]

]

d c

ba

[01 02]

[01 04][01 04]

[01 02]

[03 05] [03 05]

[03 05][03 05]

G

Figure 1 119866 is regular and totally regular

[[[

[

119886119887 119887119888 119888119889 119889119886

120583minus119861

01 01 01 01

120583+119861

02 04 02 04

]]]

]

(3)

Routine computations show that an interval-valued fuzzygraph 119866 as shown in Figure 1 is both regular and totallyregular

Example 8 Consider a graph 119866lowast such that 119881 = V

1 V2 V3

119864 = V1V2 V1V3 Let 119860 be an interval-valued fuzzy subset of

119881 and let 119861 be an interval-valued fuzzy subset of 119864 defined by

120583minus

119860(V1) = 04 120583

minus

119860(V2) = 07 120583

minus

119860(V3) = 06

120583+

119860(V1) = 04 120583

+

119860(V2) = 08 120583

+

119860(V3) = 07

120583minus

119861(V1V2) = 02 120583

minus

119861(V1V3) = 02 120583

+

119861(V1V2) = 03

120583+

119861(V1V3) = 04

(4)

Routine computations show that an interval-valued fuzzygraph 119866 is neither totally regular nor regular

Definition 9 We define the order 119874(119866) and size 119878(119866) of aninterval-valued fuzzy graph 119866 = (119860 119861) by

119874 (119866) = sum119909isin119881

1 + 120583+119860(119909) minus 120583minus

119860(119909)

2

119878 (119866) = sum119909119910isin119864

1 + 120583+119861(119909119910) minus 120583minus

119861(119909119910)

2

(5)

4 Journal of Applied Mathematics

Definition 10 An interval-valued fuzzy graph 119866 = (119860 119861) iscalled complete if

120583minus

119861(119909119910) = min (120583

minus

119860(119909) 120583

minus

119860(119910))

120583+

119861(119909119910) = max (120583

+

119860(119909) 120583

+

119860(119910)) forall119909 119910 isin 119881

(6)

Example 11 Consider a graph119866lowast such that119881 = 119909 119910 119911 119864 =

119909119910 119910119911 119911119909 Let119860 be an interval-valued fuzzy subset of119881 andlet 119861 be an interval-valued fuzzy subset of 119864 defined by

120583minus

119860(119909) = 03 120583

minus

119860(119910) = 04 120583

minus

119860(119911) = 05

120583+

119860(119909) = 05 120583

+

119860(119910) = 07 120583

+

119860(119911) = 06

120583minus

119861(119909119910) = 03 120583

minus

119861(119910119911) = 04 120583

minus

119861(119911119909) = 03

120583+

119861(119909119910) = 05 120583

+

119861(119910119911) = 06 120583

+

119861(119911119909) = 05

(7)

Routine computations show that 119866 is both completeand totally regular interval-valued fuzzy graph but 119866 is notregular since deg(119909) = deg(119911) = deg(119910)

Theorem 12 Every complete interval-valued fuzzy graph istotally regular

Theorem 13 Let119866 = (119860 119861) be an interval-valued fuzzy graphof a graph 119866

lowast Then119860 = [120583minus119860 120583+119860] is a constant function if and

only if the following statements are equivalent

(a) 119866 is a regular interval-valued fuzzy graph(b) 119866 is a totally regular interval-valued fuzzy graph

Proof Suppose that 119860 = [120583minus119860 120583+119860] is a constant function Let

120583minus119860(119909) = 119888

1and 120583+

119860(119909) = 119888

2for all 119909 isin 119881

(a) rArr (b) Assume that 119866 is an 119899-regular interval-valuedfuzzy graph Then deg

120583minus

(119909) = 1198991and deg

120583+

(119909) = 1198992for all

119909 isin 119881 So

deg120583minus[119909] = deg

120583minus(119909) + 120583

minus

119860(119909)

deg120583+[119909] = deg

120583+(119909) + 120583

+

119860(119909) forall119909 isin 119881

(8)

Thus

deg120583minus[119909] = 119899

1+ 1198881 deg

120583+[119909] = 119899

2+ 1198882 forall119909 isin 119881 (9)

Hence 119866 is a totally regular interval-valued fuzzy graph(b) rArr (a) Suppose that 119866 is a totally regular interval-

valued fuzzy graph Then

deg120583minus[119909] = 119896

1 deg

120583+[119909] = 119896

2 forall119909 isin 119881 (10)

or

deg120583minus(119909) + 120583

minus

119860(119909) = 119896

1

deg120583+(119909) + 120583

+

119860(119909) = 119896

2 forall119909 isin 119881

(11)

or

deg120583minus(119909) + 119888

1= 1198961 deg

120583+(119909) + 119888

2= 1198962 forall119909 isin 119881 (12)

or

deg120583minus(119909) = 119896

1minus 1198881 deg

120583+(119909) = 119896

2minus 1198882 forall119909 isin 119881 (13)

Thus 119866 is a regular interval-valued fuzzy graph Hence (a)and (b) are equivalent

The converse part is obvious

Theorem 14 Let 119866 be an interval-valued fuzzy graph where acrisp graph 119866lowast is an odd cycle Then 119866 is a regular interval-valued fuzzy graph if and only if 119861 is a constant function

Proof If 119861 = [120583minus119861 120583+119861] is a constant function say 120583minus

119861= 1198881and

120583+119861

= 1198882for all 119909119910 isin 119864 then deg

120583minus

(119909) = 21198881and deg

120583+

(119909) =

21198882for every119909 isin 119881 Hence119866 is a regular interval-valued fuzzy

graphConversely suppose that 119866 is a (119896

1 1198962)-regular interval-

valued fuzzy graph Let 1198901 1198902 119890

2119899+1be the edges of 119866 in

that order Let 120583minus119861(1198901) = 1198881 120583minus119861(1198902) = 119896

1minus 1198881 120583minus119861(1198903) = 119896

1minus

(1198961minus 1198881) = 1198881 120583minus119861(1198904) = 1198961minus 1198881 and so on Therefore

120583minus

119861(119890119894) =

1198881 if 119894 is odd

1198961minus 1198881 if 119894 is even

(14)

Thus 120583minus119861(1198901) = 120583minus119861(1198902119899+1

) = 1198881 So if 119890

1and 1198902119899+1

incident at avertex V

1 then deg

120583minus

(V1) = 1198961 deg120583minus

(1198901) + deg

120583minus

(1198902119899+1

) = 1198961

1198881+ 1198881= 1198961 21198881= 1198961 and 119888

1= 11989612 This shows that 120583minus

119861is a

regular functionSimilarly let 120583+

119861(1198901) = 1198882 120583+119861(1198902) = 1198962minus 1198882 120583+119861(1198903) = 1198962minus

(1198962minus 1198882) = 1198882 120583+119861(1198904) = 1198962minus 1198882 and so on Therefore

120583+

119861(119890119894) =

1198882 if 119894 is odd

1198962minus 1198882 if 119894 is even

(15)

Thus 120583+119861(1198902) = 120583+

119861(1198902119899+1

) = 1198882 So if 119890

2and 1198902119899

incident ata vertex V

2 then deg

120583+

(V2) = 1198962 deg120583+

(1198902) + deg

120583+

(1198902119899+1

) =

1198962 1198882+ 1198882

= 1198962 21198882

= 1198962 and 119888

2= 11989622 This shows that

120583+119861is a constant function Hence 119861 = [120583minus

119861 120583+119861] is a constant

function

We state the following characterization without its proof

Theorem 15 Let 119866 be an interval-valued fuzzy graph where acrisp graph 119866lowast is an even cycle Then 119866 is a regular interval-valued fuzzy graph if and only if either 119861 = [120583minus

119861 120583+119861] is a

constant function or alternate edges have the samemembershipvalues

Definition 16 The density of an interval-valued fuzzygraphs 119866 is 119863(119866) = (119863

minus(119866) 119863+(119866)) where 119863minus(119866) =(2sum119909119910isin119881

(120583minus119861(119909119910)))(sum

119909119910isin119881(120583minus119860(119909) and 120583minus

119860(119910))) for 119909 119910 isin 119881

and119863+(119866) = (2sum119909119910isin119881

(120583+119861(119909119910)))(sum

119909119910isin119881(120583+119860(119909)and120583+

119860(119910))) for

119909 119910 isin 119881 An interval-valued fuzzy graph 119866 is balanced if119863(119867) le 119863(119866) that is 119863minus(119867) le 119863minus(119866) 119863+(119867) le 119863+(119866)

for all subgraphs119867 of 119866 An interval-valued fuzzy graph 119866 isstrictly balanced if for every 119909 119910 isin 119881 119863(119867) = 119863(119866) for allnonempty subgraphs

Journal of Applied Mathematics 5

Example 17 Consider the regular interval-valued fuzzy graph119866which is given in Example 7 Routine calculations show that119863minus(119866) = 067 and 119863+(119866) = 12 Thus 119863(119866) = (067 12)

Consider that 1198671

= 119886 119887 119888 1198672

= 119886 119888 1198673

= 119886 119889 and1198674

= 119887 119888 are a nonempty subgraphs of 119866 Then 119863(1198671) =

(067 12) 119863(1198672) = (0 0) 119863(119867

3) = (067 16) and 119863(119867

4) =

(067 16) It is easy to see that regular interval-valued fuzzygraph is not balanced

Remark 18 Every regular interval-valued fuzzy graph maynot be balanced

Example 19 Consider the regular interval-valued fuzzygraph 119866 which is given in Example 11 Routine calculationsshow that 119863minus(119866) = 2 and 119863+(119866) = 2 Thus 119863(119866) = (2 2)Consider119867

1= 119909 119910119867

2= 119909 119911119867

3= 119910 119911 be a nonempty

subgraphs of 119866 Then 119863(1198671) = (2 2) 119863(119867

2) = (2 2) and

119863(1198673) = (2 2) It is easy to see that complete interval-valued

fuzzy graph is balanced 119866 is also strictly balanced

Proposition 20 Any complete interval-valued fuzzy graph isbalanced

Proposition 21 Let 119866 be a self-complementary interval-valued fuzzy graph Then 119863(119866) = (1 1)

Proposition 22 Let 1198661and 119866

2be two balanced interval-

valued fuzzy graphs Then 1198661times 1198662is balanced if and only if

119863(1198661) = 119863(119866

2) = 119863(119866

1times 1198662)

Theorem23 Let119866 be a strictly balanced interval-valued fuzzygraph and let119866 be its complement then119863(119866)+119863(119866) = (2 2)

Proof Let 119866 be a strictly balanced interval-valued fuzzygraph and 119866 its complement Let119867 be a nonempty subgraphof 119866 Since 119866 is strictly balanced 119863(119866) = 119863(119867) for every119867 sube 119866 and 119909 119910 isin 119881 In 119866

120583minus119861(119909119910) = 120583

minus

119860(119909) and 120583

minus

119860(119910) minus 120583

minus

119861(119909119910) (16)

120583+119861(119909119910) = 120583

+

119860(119909) and 120583

+

119860(119910) minus 120583

+

119861(119909119910) (17)

for every 119909 119910 isin 119881 Dividing (16) by 120583minus119860(119909) and 120583minus

119860(119910) we get

120583minus119861(119909119910)

120583minus119860(119909) and 120583minus

119860(119910)

= 1 minus120583minus119861(119909119910)

120583minus119860(119909) and 120583minus

119860(119910)

for every 119909 119910isin119881

(18)

and dividing (17) by 120583+119860(119909) and 120583+

119860(119910) we get

120583+119861(119909119910)

120583+119860(119909) and 120583+

119860(119910)

= 1 minus120583+119861(119909119910)

120583+119860(119909) and 120583+

119860(119910)

for every 119909 119910isin119881

(19)

Then

sum119909119910isin119881

120583minus119861(119909119910)

120583minus119860(119909) and 120583minus

119860(119910)

= 1 minus sum119909119910isin119881

120583minus119861(119909119910)

120583minus119860(119909) and 120583minus

119860(119910)

for every 119909 119910 isin 119881

sum119909119910isin119881

120583+119861(119909119910)

120583+119860(119909) and 120583+

119860(119910)

= 1 minus sum119909119910isin119881

120583+119861(119909119910)

120583+119860(119909) and 120583+

119860(119910)

for every 119909 119910 isin 119881

(20)

Multiplying both sides the above equations by 2

2 sum119909119910isin119881

120583minus119861(119909119910)

120583minus119860(119909) and 120583minus

119860(119910)

= 2 minus 2 sum119909119910isin119881

120583minus119861(119909119910)

120583minus119860(119909) and 120583minus

119860(119910)

for every 119909 119910 isin 119881

2 sum119909119910isin119881

120583+119861(119909119910)

120583+119860(119909) and 120583+

119860(119910)

= 2 minus 2 sum119909119910isin119881

120583+119861(119909119910)

120583+119860(119909) and 120583+

119860(119910)

for every 119909 119910 isin 119881

(21)

Thus 119863minus(119866) = 2 minus 119863minus(119866) and 119863+(119866) = 2 minus 119863+(119866)Now

119863 (119866) + 119863 (119866) = (119863minus(119866) 119863

+(119866)) + (119863

minus(119866) 119863

+(119866))

= (119863minus(119866) + 119863

minus(119866)) (119863

+(119866) + 119863

+(119866))

= (2 2)

(22)

This completes the proof

Corollary 24 The complement of strictly balanced interval-valued fuzzy graph is strictly balanced

Theorem 25 Let 1198661and 119866

2be isomorphic interval-valued

fuzzy graphs If 1198662is balanced then 119866

1is balanced

4 Irregularity in Interval-ValuedFuzzy Graphs

Definition 26 Let119866 be an interval-valued fuzzy graph on119866lowastIf there is a vertex which is adjacent to vertices with distinctneighbourhood degrees then119866 is called an irregular interval-valued fuzzy graph That is deg(119909) = 119899 for all 119909 isin 119881

6 Journal of Applied Mathematics

1

23

[01 02] [01 02]

[02 03]

[02 06]

[03 04] [02 07]

Figure 2 Irregular interval-valued fuzzy graph

Example 27 Consider a graph 119866lowast such that

119881 = V1 V2 V3 119864 = V

1V2 V2V3 V1V3 (23)

Let 119860 be an interval-valued fuzzy subset of119881 and let 119861 be aninterval-valued fuzzy subset of 119864 sube 119881 times 119881 defined by

[[[

[

V1

V2

V3

120583minus119860

02 02 03

120583+119860

06 07 04

]]]

]

[[[

[

V1V2V1V3V2V3

120583minus119861

01 01 02

120583+119861

02 02 03

]]]

]

(24)

By routine computations we have deg(V1) = [05 11]

deg(V2) = [05 10] and deg(V

3) = [04 13] It is clear that

119866 as shown in Figure 2 is an irregular interval-valued fuzzygraph

Definition 28 Let 119866 be an interval-valued fuzzy graph Ifthere is a vertex which is adjacent to vertices with distinctclosed neighbourhood degrees then 119866 is called a totallyirregular interval-valued fuzzy graph

Example 29 Consider an interval-valued fuzzy graph119866 suchthat

119881 = V1 V2 V3 V4 V5

119864 = V1V2 V2V3 V2V4 V3V1 V3V4 V4V1 V4V5

(25)

By routine computations we have deg[V1] = [14 24]

deg[V2] = [14 24] deg[V

3] = [14 24] deg[V

4] = [16 26]

and deg[V5] = [06 08] It is clear from calculations that 119866 as

1 2

34

[04 06]

[04 06]

[02 02]

[03 05]

[03 07]

[01 02]

[02 02][01 04]

[0204]

[01 03]

[02 03]

[03 05]

5

Figure 3 119866 is totally irregular

1 2

4 3

[02 06]

[05 05]

[03 07]

[01 04] [01 04]

[04 04]

[01 02]

[01 02]

Figure 4 119866 is neighbourly irregular

shown in Figure 3 is a totally irregular interval-valued fuzzygraph

Definition 30 A connected interval-valued fuzzy graph 119866 issaid to be neighbourly irregular if every two adjacent verticesof 119866 have distinct open neighbourhood degree

Example 31 Consider an interval-valued fuzzy graph 119866 suchthat

119881 = V1 V2 V3 V4

119864 = V1V2 V2V3 V3V4 V4V1

(26)

By routine computations we have deg(V1) = [08 12]

deg(V2) = [06 10] deg(V

3) = [08 12] and deg(V

4) =

[06 10] Hence 119866 as shown in Figure 4 is neighbourlyirregular

Definition 32 A connected interval-valued fuzzy graph 119866 issaid to be neighbourly totally irregular if every two adjacentvertices of 119866 have distinct closed neighbourhood degree

Journal of Applied Mathematics 7

1 2

4 3

[03 06] [04 05]

[04 05] [02 07]

[01 02]

[01 02]

[01 04] [01 04]

Figure 5 119866 is neighbourly totally irregular

Example 33 Consider an interval-valued fuzzy graph119866 suchthat

119881 = V1 V2 V3 V4

119864 = V1V2 V2V3 V3V4 V4V1

(27)

By routine computations we have deg[V1] = [11 16]

deg[V2] = [09 18] deg[V

3] = [10 17] and deg[V

4] =

[09 18] Hence119866 as shown in Figure 5 is neighbourly totallyirregular

Definition 34 Let 119866 be a connected interval-valued fuzzygraph 119866 is called highly irregular if every vertex of 119866 isadjacent to vertices with distinct neighbourhood degrees

Example 35 Consider an interval-valued fuzzy graph119866 suchthat

119881 = V1 V2 V3 V4 V5 V6

119864 = V1V2 V2V3 V2V6 V3V4 V3V5 V4V5 V5V1

(28)

By routine computations we have deg(V1) = [04 08]

deg(V2) = [06 17] deg(V

3) = [09 13] deg(V

4) = [06 11]

deg(V5) = [10 18] and deg(V

6) = [01 04] Clearly 119866 as

shown in Figure 6 is highly irregular

Example 36 Consider an interval-valued fuzzy graph119866 suchthat

119881 = V1 V2 V3 V4

119864 = V1V2 V2V3 V3V4 V4V1

(29)

By routine computations we have deg(V1) = [06 10]

deg(V2) = [08 09] deg(V

3) = [06 10] and deg(V

4) =

1 2

4

35

[02 02]

[01 03]

[03 04]

[02

04

][02

04]

[05 05]

[02 06] [01 04]

[01 04]

[03 07]

[01 0

4]

[02 02]

[02 03]

6

Figure 6 119866 is highly irregular

[08 09]We see that every two adjacent vertices have distinctopen neighbourhood degree But the vertex V

2adjacent to

the vertices V1and V

3has the same neighbourhood degree

that is deg(V1) = deg(V

3) Hence 119866 as shown in Figure 7 is

neighbourly irregular but not highly irregular

Remark 37 A neighbourly irregular interval-valued fuzzygraph may not be highly irregular

Theorem38 An interval-valued fuzzy graph119866 is highly irreg-ular and neighbourly irregular interval-valued fuzzy graph ifand only if the neighbourhood degrees of all the vertices of 119866are distinct

Proof Let 119866 be an interval-valued fuzzy graph with 119899-vertices V

1 V2 V

119899 Assume that 119866 is highly irregular and

neighbourly irregular

Claim 1 The neighbourhood degrees of all vertices of 119866

are distinct Let deg(V119894) = [119896

119894 119897119894] 119894 = 1 2 119899 Let

the adjacent vertices of V1be V2 V3 V

119899with neighbour-

hood degrees [1198962 1198972] [1198963 1198973] [119896

119899 119897119899] respectively Then

1198962

= 1198963

= sdot sdot sdot = 119896119899

and 1198972

= 1198973

= sdot sdot sdot = 119897119899 since 119866 is highly

irregular Also 1198961

= 1198962

= 1198963

= sdot sdot sdot = 119896119899and 1198971

= 1198972

= 1198973

= sdot sdot sdot = 119897119899

since 119866 is neighbourly irregular Hence the neighbourhooddegree of all the vertices of 119866 is distinct

8 Journal of Applied Mathematics

1 2

34

[03 04]

[01 04]

[04 05]

[01 02]

[01 02]

[02 05]

[05 05]

[01 04]

Figure 7 119866 is neighbourly irregular but not highly irregular

Conversely assume that the neighbourhood degrees of allthe vertices of 119866 are distinct

Claim 2 119866 is highly irregular and neighbourly irregularinterval-valued fuzzy graph

Let deg(V119894) = [119896

119894 119897119894] 119894 = 1 2 119899 Given that

1198961

= 1198962

= 1198963

= sdot sdot sdot = 119896119899and 1198971

= 1198972

= 1198973

= sdot sdot sdot = 119897119899 which implies

that every two adjacent vertices have distinct neighbourhooddegrees and to every vertex the adjacent vertices have distinctneighbourhood degrees

Theorem 39 An interval-valued fuzzy graph 119866 of 119866lowast where119866lowast is a cycle with 3 vertices that is neighbourly irregular andhighly irregular if and only if the lower and upper membershipvalues of the vertices between every pair of vertices are alldistinct

Proof Assume that lower and upper membership values ofthe vertices are all distinct

Claim 1 119866 is neighbourly irregular and highly irregularinterval-valued fuzzy graph

Let V119894 V119895 V119896

isin 119881 Given that 120583minus119860(V119894) = 120583minus119860(V119895) =

120583minus

119860(V119896) and 120583

+

119860(V119894) = 120583+

119860(V119895) = 120583+

119860(V119896) which implies that

sum119909isin119873(119909)

120583minus119860(V119894) = sum119909isin119873(119909)

120583minus119860(V119895) = sum119909isin119873(119909)

120583minus119860(V119896) and

sum119909isin119873(119909)

120583+119860(V119894) = sum119909isin119873(119909)

120583+119860(V119895) = sum119909isin119873(119909)

120583+119860(V119896) That is

deg(V119894) = deg(V

119895) = deg(V

119896) Hence 119866 is neighbourly

irregular and highly irregularConversely assume that 119866 is neighbourly irregular and

highly irregular

Claim 2 Lower and upper membership values of the verticesare all distinct

Let deg(V119894) = [119896

119894 119897119894] 119894 = 1 2 119899 Suppose that lower

and upper membership value of any two vertices are the

1

23

[04 06]

[02 06]

[02 07] [02 06]

[04 06]

[04 06]

Figure 8 119866 is not neighbourly irregular but it is complete

same Let V1 V2

isin 119881 Let 120583minus119860(V1) = 120583minus

119860(V2) and 120583+

119860(V1) =

120583+119860(V2) Then deg(V

1) = deg(V

2) since 119866lowast is cycle which is

a contradiction to the fact that119866 is neighbourly irregular andhighly irregular interval-valued fuzzy graph Hence lowermembership and upper membership value of the vertices areall distinct

Remark 40 A complete interval-valued fuzzy graph may notbe neighbourly irregular

Example 41 Consider an interval-valued fuzzy graph 119866 suchthat

119881 = V1 V2 V3 119864 = V

1V2 V2V3 V1V3 (30)

By routine computations we have deg(V1) = [06 13]

deg(V2) = [06 13] and deg(V

3) = [08 12] We see that

neighbourhood degree of V1and V

2is not distinct Hence

119866 as shown in Figure 8 is not neighbourly irregular but it iscomplete

Remark 42 A neighbourly total irregular interval-valuedfuzzy graph may not be neighbourly irregular

Example 43 Consider an interval-valued fuzzy graph119866 suchthat

119881 = V1 V2 V3 V4 119864 = V

1V2 V2V3 V3V4 V4V1 (31)

By routine computations we have deg(V1) = [07 09]

deg(V2) = [08 11] deg(V

3) = [07 09] deg(V

4) = [08 11]

deg[V1] = [11 15] deg[V

2] = [11 15] deg[V

3] = [11 14]

and deg[V4] = [12 16] We see that deg[V

1] = deg[V

2]

Hence 119866 as shown in Figure 9 is neighbourly irregular butnot a neighbourly total irregular

Journal of Applied Mathematics 9

1 2

34

[04 06]

[01 04]

[04 05] [01 02]

[01 02]

[04 05]

[03 04]

[01 04]

Figure 9 119866 is neighbourly irregular but not neighbourly totalirregular

Proposition 44 If an interval-valued fuzzy graph 119866 is neigh-bourly irregular and [120583minus

119860 120583+119860] is a constant function then it is

neighbourly totally irregular

Proof Assume that 119866 is a neighbourly irregular interval-valued fuzzy graph Then the open neighbourhood degreesof every two adjacent vertices are distinct Let V

119894 V119895

isin 119881 beadjacent vertices with distinct open neighbourhood degrees[1198961 1198971] and [119896

2 1198972] where 119896

1= 1198962 1198971

= 1198972 Let us assume that

(1205831(V119894) ]1(V119894)) = (120583

1(V119895) ]1(V119895)) = [119888

1 1198882] where 119888

1 1198882are

constant and 1198881 1198882isin [0 1] Therefore deg

120583minus

[V119894] = deg

120583minus

(V119894) +

1205831(V119894) = 119896

1+ 1198881 deg120583+

[V119894] = deg

120583+

(V119894) + ]1(V119894) = 119897

1+

1198882deg120583minus

[V119895] = deg

120583minus

(V119895) + 1205831(V119895) = 1198962+ 1198881 and deg

120583+

[V119895] =

deg120583+

(V119895) + ]1(V119895) = 1198972+ 1198882

Claim Consider that deg120583minus

[V119894] = deg

120583minus

[V119895] and deg

120583+

[V119894] =

deg120583+

[V119895] Suppose that deg

120583minus

[V119894] = deg

120583minus

[V119895] and

deg120583+

[V119894] = deg

120583+

[V119895] Consider that

deg120583minus

[V119894] = deg

120583minus

[V119895]

1198961+ 1198881= 1198962+ 1198881

1198961minus 1198962= 1198881minus 1198881= 0

1198961= 1198962 which is a contradiction to 119896

1= 1198962

(32)

Therefore deg120583minus

[V119894] = deg

120583minus

[V119895] Similarly we consider that

deg120583+

[V119894] = deg

120583+

[V119895]

1198971+ 1198882= 1198972+ 1198882

1198971minus 1198972= 1198882minus 1198882= 0

1198971= 1198972 which is a contradiction to 119897

1= 1198972

(33)

Therefore deg120583+

[V119894] = deg

120583+

[V119895] Hence 119866 is a neighbourly

totally irregular interval-valued fuzzy graph

Theorem 45 If an interval-valued fuzzy graph 119866 is neigh-bourly totally irregular and [120583minus

119860 120583+119860] is a constant function

then it is a neighbourly irregular interval-valued fuzzy graph

Proof Assume that119866 is a neighbourly total irregular interval-valued fuzzy graph Then the closed neighbourhood degreeof every two adjacent vertices is distinct Let V

119894 V119895

isin 119881

and deg[V119894] = [119896

1 1198971] deg[V

119895] = [119896

2 1198972] where 119896

1= 1198962

and 1198971

= 1198972 Assume that (120583

1(V119894) ]1(V119894)) = [119888

1 1198882] and

(1205831(V119895) ]1(V119895)) = [119888

1 1198882] where 119888

1 1198882

isin [0 1] are constantand deg[V

119894] = deg[V

119895]

Claim Consider that deg(V119894) = deg(V

119895)

Given that deg[V119894] = deg[V

119895] which implies deg

120583minus

[V119894] =

deg120583minus

[V119895] and deg

120583+

[V119894] = deg

120583+

[V119895] now we consider that

deg120583minus

[V119894] = deg

120583minus

[V119895]

1198961+ 1198881

= 1198962+ 1198881

1198961

= 1198962

(34)

We now consider that

deg120583+

[V119894] = deg

120583+

[V119895]

1198971+ 1198882

= 1198972+ 1198882

1198971

= 1198972

(35)

That is the neighbourhood degrees of adjacent vertices of 119866are distinct Hence neighbourhood degree of every pair ofadjacent vertices is distinct in 119866

Proposition 46 If an interval-valued fuzzy graph 119866 is neigh-bourly irregular and neighbourly totally irregular then [120583

minus

119860 120583+

119860]

need not be a constant function

Remark 47 If 119866 is a neighbourly irregular interval-valuedfuzzy graph then interval-valued subgraph 119867 = (119860

1015840 1198611015840) of119866 may not be neighbourly irregular

Remark 48 If 119866 is a totally irregular interval-valued fuzzygraph then interval-valued fuzzy subgraph 119867 = (1198601015840 1198611015840) of119866 may not be totally irregular

5 Relationship between IFGs and IVFGs

In 2003 Deschrijver and Kerre [31] established the rela-tionships between some extensions of fuzzy sets In thissection we present the relationship between extensions offuzzy graphs Shannon and Atanassov [32] introduced thenotion of an intuitionistic fuzzy graph Some operations onintuitionistic fuzzy graphs are discussed in [33]

Definition 49 (see [32]) By an intuitionistic fuzzy graph (IFGfor short) 119866 of a graph 119866

lowast we mean a pair 119866 = (119860 119861)

10 Journal of Applied Mathematics

Fuzzy sets

Interval-valued

fuzzy sets

Intuitionisticfuzzy sets

Interval-valued

fuzzy graphs

Intuitionisticfuzzy graphs

Figure 10 Links between models

Interval-valued

fuzzy graphs

Intuitionistic

fuzzy graphs

Fuzzy graphs

Figure 11 Link between IFGs and IVFGs

where 119860 = (120583119860 ]119860) is an intuitionistic fuzzy set on 119881 and

119861 = (120583119861 ]119861) is an intuitionistic fuzzy relation on 119864 such that

120583119861(119909119910) le min (120583

119860(119909) 120583

119860(119910))

]119861(119909119910) le max (]

119860(119909) ]

119860(119910))

(36)

for all 119909119910 isin 119864 The class of all IFGs on 119866lowast will be denoted byIFG(119866lowast)

Theorem 50 (see [33]) If 1198661and 119866

2are intuitionistic fuzzy

graphs then 1198661cap 1198662 1198661cup 1198662 and 119866

1are intuitionistic fuzzy

graphs

Ju and Wang introduced the notion of interval-valuedfuzzy graph (IVFG for short) in [15] Some operations oninterval-valued fuzzy graphs are discussed in [18] The classof all IVFGs on 119866

lowast will be denoted byIVFG(119866lowast)

Theorem 51 (see [19]) If 1198661and 119866

2are interval-valued fuzzy

graphs then1198661cap11986621198661cup1198662 and119866

1are interval-valued fuzzy

graphs

Lemma 52 (IVFG(119866lowast) cup cap) and (IFG(119866lowast) cup cap) arecomplete lattices

Theorem 53 The mapping

1205951 IVFG (119866

lowast) 997888rarr IFG (119866

lowast) (37)

defined by

1205951(119861) = ((119909 120583

minus

119860(119909) 1 minus 120583

+

119860(119909)) | 119909 isin 119881

(119909119910 120583minus

119861(119909119910) 1 minus 120583

+

119861(119909119910)) | 119909119910 isin 119864)

(38)

where119861 = ((119909 120583

minus

119860(119909) 120583

+

119860(119909)) | 119909 isin 119881

(119909119910 120583minus

119861(119909119910) 120583

+

119861(119909119910)) | 119909119910 isin 119864)

(39)

is an isomorphism between lattices (IVFG(119866lowast) cup cap) and(IFG(119866lowast) cup cap)

Remark 54 From a pure mathematical point of viewTheorem 53 shows that the two concepts intuitionistic fuzzygraphs and interval-valued fuzzy graphs are equivalent

In Figures 10 and 11 we present the relationships that existbetween different models In these figures a double arrowbetween two theories means that they are equivalent a singlearrow 119883 rarr 119884 denotes that 119884 is an extension of 119883 InFigure 10 a dash arrow 119883[119884 denotes that model 119884 is basedon the previous model 119883

6 Conclusions

Graph theory has several interesting applications in systemanalysis operations research computer applications andeconomics Since most of the time the aspects of graphproblems are uncertain it is nice to deal with these aspectsvia the methods of fuzzy systems It is known that fuzzygraph theory has numerous applications in modern scienceand engineering especially in the field of information theoryneural networks expert systems cluster analysis medicaldiagnosis traffic engineering network routing town plan-ning and control theory Since interval-valued fuzzy settheory is an increasingly popular extension of fuzzy settheory where traditional [0 1]-valued membership degreesare replaced by intervals in [0 1] that approximate the(unknown) membership degrees specific types of interval-valued fuzzy graphs have been introduced and investigatedin this paper The natural extension of this research work isthe application of interval-valued fuzzy graphs in the area ofsoft computing including neural networks expert systemsdatabase theory and geographical information systems

Acknowledgments

The authors are thankful to the referees for their valuablecomments The authors are also thankful to Professor SyedMansoor Sarwar and Dr Faisal Aslam for their valuablesuggestions

References

[1] S G Shirinivas S Vetrivel and N M Elango ldquoApplications ofgraph theory in computer sciencemdashan overviewrdquo InternationalJournal of Engineering Science and Technology vol 2 no 9 pp4610ndash4621 2010

Journal of Applied Mathematics 11

[2] N Deo Graph Theory with Applications to Engineering andComputer Science Prentice Hall Englewood Cliffs NJ USA1990

[3] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoningmdashIrdquo Information Sciencesvol 8 pp 199ndash249 1975

[4] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[5] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 pp338ndash353 1965

[6] J M Mendel Uncertain Rule-Based Fuzzy Logic Systems Intro-duction and New Directions Prentice-Hall Upper Saddle RiverNJ USA 2001

[7] M B Gorzałczany ldquoA method of inference in approximatereasoning based on interval-valued fuzzy setsrdquo Fuzzy Sets andSystems vol 21 no 1 pp 1ndash17 1987

[8] M B Gorzałczany ldquoAn interval-valued fuzzy inferencemethodmdashsome basic propertiesrdquo Fuzzy Sets and Systems vol31 no 2 pp 243ndash251 1989

[9] M K Roy and R Biswas ldquoI-V fuzzy relations and Sanchezrsquosapproach for medical diagnosisrdquo Fuzzy Sets and Systems vol47 no 1 pp 35ndash38 1992

[10] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986

[11] A Kauffman Introduction a la Theorie des Sous-emsemblesFlous vol 1 Masson et Cie 1973

[12] A Rosenfeld ldquoFuzzy graphsrdquo in Fuzzy Sets and Their Applica-tions L A Zadeh K S Fu and M Shimura Eds pp 77ndash95Academic Press New York NY USA 1975

[13] J N Mordeson and C-S Peng ldquoOperations on fuzzy graphsrdquoInformation Sciences vol 79 no 3-4 pp 159ndash170 1994

[14] M S Sunitha and A V Kumar ldquoComplement of a fuzzy graphrdquoIndian Journal of Pure and Applied Mathematics vol 33 no 9pp 1451ndash1464 2002

[15] H Ju and L Wang ldquoInterval-valued fuzzy subsemigroupsand subgroups associated by intervalvalued fuzzy graphsrdquo inProceedings of the WRI Global Congress on Intelligent Systems(GCIS rsquo09) pp 484ndash487 Xiamen China May 2009

[16] MAkram ldquoBipolar fuzzy graphsrdquo Information Sciences vol 181no 24 pp 5548ndash5564 2011

[17] M Akram ldquoBipolar fuzzy graphs with applicationsrdquo KnowledgeBased Systems vol 39 pp 1ndash8 2013

[18] M Akram ldquoInterval-valued fuzzy line graphsrdquoNeural Comput-ing and Applications vol 21 pp 145ndash150 2012

[19] M Akram and W A Dudek ldquoInterval-valued fuzzy graphsrdquoComputers amp Mathematics with Applications vol 61 no 2 pp289ndash299 2011

[20] M Akram and B Davvaz ldquoStrong intuitionistic fuzzy graphsrdquoFilomat vol 26 no 1 pp 177ndash196 2012

[21] M Akram K H Dar and K P Shum ldquoInterval-valued (120572120573)-fuzzy K-algebrasrdquo Applied Soft Computing Journal vol 11 no 1pp 1213ndash1222 2011

[22] T AL-Hawary ldquoComplete fuzzy graphsrdquo International Journalof Mathematical Combinatorics vol 4 pp 26ndash34 2011

[23] P Bhattacharya ldquoSome remarks on fuzzy graphsrdquo PatternRecognition Letters vol 6 no 5 pp 297ndash302 1987

[24] S M Chen ldquoInterval-valued fuzzy hypergraph and fuzzypartitionrdquo IEEE Transactions on Systems Man and CyberneticsB vol 27 no 4 pp 725ndash733 1997

[25] G Deschrijver and C Cornelis ldquoRepresentability in interval-valued fuzzy set theoryrdquo International Journal of UncertaintyFuzziness and Knowledge-Based Systems vol 15 no 3 pp 345ndash361 2007

[26] A N Gani and S R Latha ldquoOn irregular fuzzy graphsrdquoAppliedMathematical Sciences vol 6 no 9ndash12 pp 517ndash523 2012

[27] X Ma J Zhan B Davvaz and Y B Jun ldquoSome kinds of (isin isinor119902)-interval-valued fuzzy ideals of BCI-algebrasrdquo InformationSciences vol 178 no 19 pp 3738ndash3754 2008

[28] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Studies in Fuzziness and SoftComputing PhysicaHeidelberg Germany 1998

[29] F Riaz and KM Ali ldquoApplications of graph theory in computersciencerdquo in Proceedings of the 3rd International Conferenceon Computational Intelligence Communication Systems andNetworks (CICSyN rsquo11) pp 142ndash145 Bali Indonesia July 2011

[30] M Behzad and G Chartrand ldquoNo graph is perfectrdquo TheAmerican Mathematical Monthly vol 74 pp 962ndash963 1967

[31] G Deschrijver and E E Kerre ldquoOn the relationship betweensome extensions of fuzzy set theoryrdquo Fuzzy Sets and Systemsvol 133 no 2 pp 227ndash235 2003

[32] A Shannon and K T Atanassov ldquoA first step to a theory ofthe intuitionistic fuzzy graphsrdquo in Proceeding of the FUBESTD Lakov Ed pp 59ndash61 Sofia Bulgaria 1994

[33] R Parvathi M G Karunambigai and K T Atanassov ldquoOper-ations on intuitionistic fuzzy graphsrdquo in Proceedings of theIEEE International Conference on Fuzzy Systems pp 1396ndash1401August 2009

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Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

differences between the traditional numerical models used inengineering and sciences and the symbolic models used inexpert systems Mordeson and Peng [13] defined the conceptof complement of fuzzy graph and described some operationson fuzzy graphs In [14] the definition of complement ofa fuzzy graph was modified so that the complement of thecomplement is the original fuzzy graph which agreeswith thecrisp graph case Ju andWang gave the definition of interval-valued fuzzy graph in [15] Akram et al [16ndash20] introducedmany new concepts including bipolar fuzzy graphs interval-valued line fuzzy graphs and strong intuitionistic fuzzygraphs In this paper we propose certain types of interval-valued fuzzy graphs including balanced interval-valued fuzzygraphs neighbourly irregular interval-valued fuzzy graphsneighbourly total irregular interval-valued fuzzy graphshighly irregular interval-valued fuzzy graphs and highlytotal irregular interval-valued fuzzy graphs Some interestingproperties associated with these new interval-valued fuzzygraphs are investigated and necessary and sufficient condi-tions under which neighbourly irregular and highly irregularinterval-valued fuzzy graphs are equivalent are obtained Wealso describe the relationship between intuitionistic fuzzygraphs and interval-valued fuzzy graphs

We used standard definitions and terminologies in thispaper For notations terminologies and applications are notmentioned in the paper the readers are referred to [13 14 21ndash29]

2 Preliminaries

In this section we review some elementary concepts whoseunderstanding is necessary to fully benefit from this paper

By a graph 119866lowast = (119881 119864) we mean a nontrivial finite

connected and undirected graph without loops or multipleedges We write 119909119910 isin 119864 to mean (119909 119910) isin 119864 and if 119890 = 119909119910 isin 119864we say that 119909 and 119910 are adjacent Formally given a graph119866lowast = (119881 119864) two vertices 119909 119910 isin 119881 are said to be neighbors

or adjacent nodes if 119909119910 isin 119864 The number of vertices thecardinality of 119881 is called the order of graph and denotedby |119881| The number of edges the cardinality of 119864 is calledthe size of graph and denoted by |119864| A path in a graph 119866

lowast

is an alternating sequence of vertices and edges V0 1198901 V1

1198902 V

119899minus1 119890119899 V119899 The path graph with 119899 vertices is denoted

by 119875119899 A path is sometimes denoted by 119875

119899 V0V1sdot sdot sdot V119899(119899 gt 0)

The length of a path 119875119899in119866lowast is 119899 A path 119875

119899 V0V1sdot sdot sdot V119899in119866lowast

is called a cycle if V0= V119899and 119899 ge 3 Note that path graph 119875

119899

has 119899 minus 1 edges and can be obtained from a cycle graph 119862119899

by removing any edge An undirected graph 119866lowast is connected

if there is a path between each pair of distinct vertices Theneighbourhood of a vertex V in a graph 119866

lowast is the inducedsubgraph of 119866lowast consisting of all vertices adjacent to V andall edges connecting two such vertices The neighbourhoodis often denoted 119873(V) The degree deg(V) of vertex V is thenumber of edges incident on V or equivalently deg(V) =

|119873(V)| The set of neighbors called a (open) neighborhood119873(V) for a vertex V in a graph 119866

lowast consists of all verticesadjacent to V but not including V that is119873(V) = 119906 isin 119881 | V119906 isin

119864 When V is also included it is called a closed neighborhood119873[V] that is 119873[V] = 119873(V) cup V A regular graph is a graph

where each vertex has the same number of neighbors that isall the vertices have the same closed neighbourhood degreeA connected graph is highly irregular if each of its vertices isadjacent only to vertices with distinct degrees Equivalentlya graph 119866

lowast is highly irregular if every two vertices of 119866lowast

connected by a path of length 2 have distinct degrees Aconnected graph is said to be neighbourly irregular if no twoadjacent vertices of 119866lowast have the same degree Equivalently aconnected graph 119866lowast is called neighbourly irregular if everytwo adjacent vertices of 119866lowast have distinct degree

It is known that one of the best known classes of graphs isthe class of regular graphs These graphs have been studiedextensively in various contexts Regular graphs of degree 119903

and order 119899 exist with only limited but natural restrictionsIndeed for integers 119903 and 119899 with 0 le 119903 le 119899 minus 1 an 119903-regulargraph of order 119899 exists if and only if 119899119903 is even A graph thatis not regular will be called irregular It is well known [30]that all nontrivial graphs regular or irregular must containat least two vertices of the same degree In a regular graphof course every vertex is adjacent only to vertices having thesame degree On the other hand it is possible for a vertex inan irregular graph to be adjacent only to vertices with distinctdegrees With these observations made we now considergraphs that are opposite in a certain sense to regular graphsWe consider only undirected graphs with the finite numberof vertices and edges

Applications of fuzzy relations arewidespread and impor-tant especially in the field of clustering analysis neuralnetworks computer networks pattern recognition decisionmaking and expert systems In each of these the basicmathematical structure is that of a fuzzy graph

Definition 1 (see [3 5]) A fuzzy subset 120583 on a set 119883 is a map120583 119883 rarr [0 1] A fuzzy binary relation on119883 is a fuzzy subset120583 on 119883 times 119883 By a fuzzy relation we mean a fuzzy binaryrelation given by 120583 119883 times 119883 rarr [0 1]

Fuzzy set theory is an extension of ordinary set theoryin which to each element a real number between 0 and 1called the membership degree is assigned Unfortunately itis not always possible to give an exact degree of membershipThere can be uncertainty about the membership degreebecause of lack of knowledge vague information and soforth A possible way to overcome this problem is to useinterval-valued fuzzy sets which assign to each element aclosed interval which approximates the ldquorealrdquo but unknownmembership degree The length of this interval is a measurefor the uncertainty about the membership degree

An interval number119863 is an interval [119886minus 119886+]with 0 le 119886minus le

119886+ le 1 The interval [119886 119886] is identified with the number 119886 isin

[0 1] Let119863[0 1] be the set of all closed subintervals of [0 1]Then it is known that (119863[0 1] le or and) is a complete latticewith [0 0] as the least element and [1 1] as the greatest

Definition 2 (see [7]) An interval-valued fuzzy relation 119877 ina universe119883times119884 is a mapping 119877 119883times119884 rarr 119863[0 1] such that119877(119909 119910) = [119877minus(119909 119910) 119877+(119909 119910)] isin 119863[0 1] for all pairs (119909 119910) isin

119883 times 119884

Journal of Applied Mathematics 3

Interval-valued fuzzy relations reflect the idea that mem-bership grades are often not precise and the intervals repre-sent such uncertainty

Definition 3 (see [15]) By an interval-valued fuzzy graph119866 ofa graph 119866

lowast we mean a pair 119866 = (119860 119861) where 119860 = [120583minus119860 120583+119860]

is an interval-valued fuzzy set on 119881 and 119861 = [120583minus119861 120583+119861] is an

interval-valued fuzzy relation on 119864 such that

120583minus

119861(119909119910) le min (120583

minus

119860(119909) 120583

minus

119860(119910))

120583+

119861(119909119910) le min (120583

+

119860(119909) 120583

+

119860(119910))

(1)

for all 119909119910 isin 119864

Throughout this paper 119866lowast is a crisp graph and 119866 is aninterval-valued fuzzy graph

3 Balanced Interval-Valued Fuzzy Graphs

Definition 4 Let 119866 be an interval-valued fuzzy graph Theneighbourhood degree of a vertex 119909 in119866 is defined by deg(119909) =

[deg120583minus

(119909) deg120583+

(119909)] where deg120583minus

(119909) = sum119910isin119873(119909)

120583minus119860(119910) and

deg120583+

(119909) = sum119910isin119873(119909)

120583+119860(119910) Notice that 120583minus

119861(119909119910) gt 0 120583+

119861(119909119910) gt

0 for 119909119910 isin 119864 and 120583minus119861(119909119910) = 120583+

119861(119909119910) = 0 for 119909119910 notin 119864

Definition 5 Let 119866 = (119860 119861) be an interval-valued fuzzygraph on119866lowast If all the vertices have the same open neighbour-hood degree 119899 then 119866 is called an 119899-regular interval-valuedfuzzy graph The open neighbourhood degree of a vertex 119909

in 119866 is defined by deg(119909) = [deg120583minus

(119909) deg120583+

(119909)] wheredeg120583minus

(119909) = sum119910isin119873(119909)

120583minus119860(119910) and deg

120583+

(119909) = sum119910isin119873(119909)

120583+119860(119910)

Definition 6 Let 119866 be an interval-valued fuzzy graph Theclosed neighbourhood degree of a vertex 119909 is defined bydeg[119909] = [deg

120583minus

[119909] deg120583+

[119909]] where

deg120583minus[119909] = deg

120583minus(119909) + 120583

minus

119860(119909)

deg120583+[119909] = deg

120583+(119909) + 120583

+

119860(119909)

(2)

If all the vertices have the same closed neighbourhood degree119898 then 119866 is called a 119898-totally regular interval-valued fuzzygraph

Example 7 Consider a graph 119866lowast such that 119881 = 119886 119887 119888 119889119864 = 119886119887 119887119888 119888119889 119886119889 Let 119860 be an interval-valued fuzzy subsetof 119881 and let119861 be an interval-valued fuzzy subset of119864 sube 119881times119881

defined by

[[[

[

119886 119887 119888 119889

120583minus119860

03 03 03 03

120583+119860

05 05 05 05

]]]

]

d c

ba

[01 02]

[01 04][01 04]

[01 02]

[03 05] [03 05]

[03 05][03 05]

G

Figure 1 119866 is regular and totally regular

[[[

[

119886119887 119887119888 119888119889 119889119886

120583minus119861

01 01 01 01

120583+119861

02 04 02 04

]]]

]

(3)

Routine computations show that an interval-valued fuzzygraph 119866 as shown in Figure 1 is both regular and totallyregular

Example 8 Consider a graph 119866lowast such that 119881 = V

1 V2 V3

119864 = V1V2 V1V3 Let 119860 be an interval-valued fuzzy subset of

119881 and let 119861 be an interval-valued fuzzy subset of 119864 defined by

120583minus

119860(V1) = 04 120583

minus

119860(V2) = 07 120583

minus

119860(V3) = 06

120583+

119860(V1) = 04 120583

+

119860(V2) = 08 120583

+

119860(V3) = 07

120583minus

119861(V1V2) = 02 120583

minus

119861(V1V3) = 02 120583

+

119861(V1V2) = 03

120583+

119861(V1V3) = 04

(4)

Routine computations show that an interval-valued fuzzygraph 119866 is neither totally regular nor regular

Definition 9 We define the order 119874(119866) and size 119878(119866) of aninterval-valued fuzzy graph 119866 = (119860 119861) by

119874 (119866) = sum119909isin119881

1 + 120583+119860(119909) minus 120583minus

119860(119909)

2

119878 (119866) = sum119909119910isin119864

1 + 120583+119861(119909119910) minus 120583minus

119861(119909119910)

2

(5)

4 Journal of Applied Mathematics

Definition 10 An interval-valued fuzzy graph 119866 = (119860 119861) iscalled complete if

120583minus

119861(119909119910) = min (120583

minus

119860(119909) 120583

minus

119860(119910))

120583+

119861(119909119910) = max (120583

+

119860(119909) 120583

+

119860(119910)) forall119909 119910 isin 119881

(6)

Example 11 Consider a graph119866lowast such that119881 = 119909 119910 119911 119864 =

119909119910 119910119911 119911119909 Let119860 be an interval-valued fuzzy subset of119881 andlet 119861 be an interval-valued fuzzy subset of 119864 defined by

120583minus

119860(119909) = 03 120583

minus

119860(119910) = 04 120583

minus

119860(119911) = 05

120583+

119860(119909) = 05 120583

+

119860(119910) = 07 120583

+

119860(119911) = 06

120583minus

119861(119909119910) = 03 120583

minus

119861(119910119911) = 04 120583

minus

119861(119911119909) = 03

120583+

119861(119909119910) = 05 120583

+

119861(119910119911) = 06 120583

+

119861(119911119909) = 05

(7)

Routine computations show that 119866 is both completeand totally regular interval-valued fuzzy graph but 119866 is notregular since deg(119909) = deg(119911) = deg(119910)

Theorem 12 Every complete interval-valued fuzzy graph istotally regular

Theorem 13 Let119866 = (119860 119861) be an interval-valued fuzzy graphof a graph 119866

lowast Then119860 = [120583minus119860 120583+119860] is a constant function if and

only if the following statements are equivalent

(a) 119866 is a regular interval-valued fuzzy graph(b) 119866 is a totally regular interval-valued fuzzy graph

Proof Suppose that 119860 = [120583minus119860 120583+119860] is a constant function Let

120583minus119860(119909) = 119888

1and 120583+

119860(119909) = 119888

2for all 119909 isin 119881

(a) rArr (b) Assume that 119866 is an 119899-regular interval-valuedfuzzy graph Then deg

120583minus

(119909) = 1198991and deg

120583+

(119909) = 1198992for all

119909 isin 119881 So

deg120583minus[119909] = deg

120583minus(119909) + 120583

minus

119860(119909)

deg120583+[119909] = deg

120583+(119909) + 120583

+

119860(119909) forall119909 isin 119881

(8)

Thus

deg120583minus[119909] = 119899

1+ 1198881 deg

120583+[119909] = 119899

2+ 1198882 forall119909 isin 119881 (9)

Hence 119866 is a totally regular interval-valued fuzzy graph(b) rArr (a) Suppose that 119866 is a totally regular interval-

valued fuzzy graph Then

deg120583minus[119909] = 119896

1 deg

120583+[119909] = 119896

2 forall119909 isin 119881 (10)

or

deg120583minus(119909) + 120583

minus

119860(119909) = 119896

1

deg120583+(119909) + 120583

+

119860(119909) = 119896

2 forall119909 isin 119881

(11)

or

deg120583minus(119909) + 119888

1= 1198961 deg

120583+(119909) + 119888

2= 1198962 forall119909 isin 119881 (12)

or

deg120583minus(119909) = 119896

1minus 1198881 deg

120583+(119909) = 119896

2minus 1198882 forall119909 isin 119881 (13)

Thus 119866 is a regular interval-valued fuzzy graph Hence (a)and (b) are equivalent

The converse part is obvious

Theorem 14 Let 119866 be an interval-valued fuzzy graph where acrisp graph 119866lowast is an odd cycle Then 119866 is a regular interval-valued fuzzy graph if and only if 119861 is a constant function

Proof If 119861 = [120583minus119861 120583+119861] is a constant function say 120583minus

119861= 1198881and

120583+119861

= 1198882for all 119909119910 isin 119864 then deg

120583minus

(119909) = 21198881and deg

120583+

(119909) =

21198882for every119909 isin 119881 Hence119866 is a regular interval-valued fuzzy

graphConversely suppose that 119866 is a (119896

1 1198962)-regular interval-

valued fuzzy graph Let 1198901 1198902 119890

2119899+1be the edges of 119866 in

that order Let 120583minus119861(1198901) = 1198881 120583minus119861(1198902) = 119896

1minus 1198881 120583minus119861(1198903) = 119896

1minus

(1198961minus 1198881) = 1198881 120583minus119861(1198904) = 1198961minus 1198881 and so on Therefore

120583minus

119861(119890119894) =

1198881 if 119894 is odd

1198961minus 1198881 if 119894 is even

(14)

Thus 120583minus119861(1198901) = 120583minus119861(1198902119899+1

) = 1198881 So if 119890

1and 1198902119899+1

incident at avertex V

1 then deg

120583minus

(V1) = 1198961 deg120583minus

(1198901) + deg

120583minus

(1198902119899+1

) = 1198961

1198881+ 1198881= 1198961 21198881= 1198961 and 119888

1= 11989612 This shows that 120583minus

119861is a

regular functionSimilarly let 120583+

119861(1198901) = 1198882 120583+119861(1198902) = 1198962minus 1198882 120583+119861(1198903) = 1198962minus

(1198962minus 1198882) = 1198882 120583+119861(1198904) = 1198962minus 1198882 and so on Therefore

120583+

119861(119890119894) =

1198882 if 119894 is odd

1198962minus 1198882 if 119894 is even

(15)

Thus 120583+119861(1198902) = 120583+

119861(1198902119899+1

) = 1198882 So if 119890

2and 1198902119899

incident ata vertex V

2 then deg

120583+

(V2) = 1198962 deg120583+

(1198902) + deg

120583+

(1198902119899+1

) =

1198962 1198882+ 1198882

= 1198962 21198882

= 1198962 and 119888

2= 11989622 This shows that

120583+119861is a constant function Hence 119861 = [120583minus

119861 120583+119861] is a constant

function

We state the following characterization without its proof

Theorem 15 Let 119866 be an interval-valued fuzzy graph where acrisp graph 119866lowast is an even cycle Then 119866 is a regular interval-valued fuzzy graph if and only if either 119861 = [120583minus

119861 120583+119861] is a

constant function or alternate edges have the samemembershipvalues

Definition 16 The density of an interval-valued fuzzygraphs 119866 is 119863(119866) = (119863

minus(119866) 119863+(119866)) where 119863minus(119866) =(2sum119909119910isin119881

(120583minus119861(119909119910)))(sum

119909119910isin119881(120583minus119860(119909) and 120583minus

119860(119910))) for 119909 119910 isin 119881

and119863+(119866) = (2sum119909119910isin119881

(120583+119861(119909119910)))(sum

119909119910isin119881(120583+119860(119909)and120583+

119860(119910))) for

119909 119910 isin 119881 An interval-valued fuzzy graph 119866 is balanced if119863(119867) le 119863(119866) that is 119863minus(119867) le 119863minus(119866) 119863+(119867) le 119863+(119866)

for all subgraphs119867 of 119866 An interval-valued fuzzy graph 119866 isstrictly balanced if for every 119909 119910 isin 119881 119863(119867) = 119863(119866) for allnonempty subgraphs

Journal of Applied Mathematics 5

Example 17 Consider the regular interval-valued fuzzy graph119866which is given in Example 7 Routine calculations show that119863minus(119866) = 067 and 119863+(119866) = 12 Thus 119863(119866) = (067 12)

Consider that 1198671

= 119886 119887 119888 1198672

= 119886 119888 1198673

= 119886 119889 and1198674

= 119887 119888 are a nonempty subgraphs of 119866 Then 119863(1198671) =

(067 12) 119863(1198672) = (0 0) 119863(119867

3) = (067 16) and 119863(119867

4) =

(067 16) It is easy to see that regular interval-valued fuzzygraph is not balanced

Remark 18 Every regular interval-valued fuzzy graph maynot be balanced

Example 19 Consider the regular interval-valued fuzzygraph 119866 which is given in Example 11 Routine calculationsshow that 119863minus(119866) = 2 and 119863+(119866) = 2 Thus 119863(119866) = (2 2)Consider119867

1= 119909 119910119867

2= 119909 119911119867

3= 119910 119911 be a nonempty

subgraphs of 119866 Then 119863(1198671) = (2 2) 119863(119867

2) = (2 2) and

119863(1198673) = (2 2) It is easy to see that complete interval-valued

fuzzy graph is balanced 119866 is also strictly balanced

Proposition 20 Any complete interval-valued fuzzy graph isbalanced

Proposition 21 Let 119866 be a self-complementary interval-valued fuzzy graph Then 119863(119866) = (1 1)

Proposition 22 Let 1198661and 119866

2be two balanced interval-

valued fuzzy graphs Then 1198661times 1198662is balanced if and only if

119863(1198661) = 119863(119866

2) = 119863(119866

1times 1198662)

Theorem23 Let119866 be a strictly balanced interval-valued fuzzygraph and let119866 be its complement then119863(119866)+119863(119866) = (2 2)

Proof Let 119866 be a strictly balanced interval-valued fuzzygraph and 119866 its complement Let119867 be a nonempty subgraphof 119866 Since 119866 is strictly balanced 119863(119866) = 119863(119867) for every119867 sube 119866 and 119909 119910 isin 119881 In 119866

120583minus119861(119909119910) = 120583

minus

119860(119909) and 120583

minus

119860(119910) minus 120583

minus

119861(119909119910) (16)

120583+119861(119909119910) = 120583

+

119860(119909) and 120583

+

119860(119910) minus 120583

+

119861(119909119910) (17)

for every 119909 119910 isin 119881 Dividing (16) by 120583minus119860(119909) and 120583minus

119860(119910) we get

120583minus119861(119909119910)

120583minus119860(119909) and 120583minus

119860(119910)

= 1 minus120583minus119861(119909119910)

120583minus119860(119909) and 120583minus

119860(119910)

for every 119909 119910isin119881

(18)

and dividing (17) by 120583+119860(119909) and 120583+

119860(119910) we get

120583+119861(119909119910)

120583+119860(119909) and 120583+

119860(119910)

= 1 minus120583+119861(119909119910)

120583+119860(119909) and 120583+

119860(119910)

for every 119909 119910isin119881

(19)

Then

sum119909119910isin119881

120583minus119861(119909119910)

120583minus119860(119909) and 120583minus

119860(119910)

= 1 minus sum119909119910isin119881

120583minus119861(119909119910)

120583minus119860(119909) and 120583minus

119860(119910)

for every 119909 119910 isin 119881

sum119909119910isin119881

120583+119861(119909119910)

120583+119860(119909) and 120583+

119860(119910)

= 1 minus sum119909119910isin119881

120583+119861(119909119910)

120583+119860(119909) and 120583+

119860(119910)

for every 119909 119910 isin 119881

(20)

Multiplying both sides the above equations by 2

2 sum119909119910isin119881

120583minus119861(119909119910)

120583minus119860(119909) and 120583minus

119860(119910)

= 2 minus 2 sum119909119910isin119881

120583minus119861(119909119910)

120583minus119860(119909) and 120583minus

119860(119910)

for every 119909 119910 isin 119881

2 sum119909119910isin119881

120583+119861(119909119910)

120583+119860(119909) and 120583+

119860(119910)

= 2 minus 2 sum119909119910isin119881

120583+119861(119909119910)

120583+119860(119909) and 120583+

119860(119910)

for every 119909 119910 isin 119881

(21)

Thus 119863minus(119866) = 2 minus 119863minus(119866) and 119863+(119866) = 2 minus 119863+(119866)Now

119863 (119866) + 119863 (119866) = (119863minus(119866) 119863

+(119866)) + (119863

minus(119866) 119863

+(119866))

= (119863minus(119866) + 119863

minus(119866)) (119863

+(119866) + 119863

+(119866))

= (2 2)

(22)

This completes the proof

Corollary 24 The complement of strictly balanced interval-valued fuzzy graph is strictly balanced

Theorem 25 Let 1198661and 119866

2be isomorphic interval-valued

fuzzy graphs If 1198662is balanced then 119866

1is balanced

4 Irregularity in Interval-ValuedFuzzy Graphs

Definition 26 Let119866 be an interval-valued fuzzy graph on119866lowastIf there is a vertex which is adjacent to vertices with distinctneighbourhood degrees then119866 is called an irregular interval-valued fuzzy graph That is deg(119909) = 119899 for all 119909 isin 119881

6 Journal of Applied Mathematics

1

23

[01 02] [01 02]

[02 03]

[02 06]

[03 04] [02 07]

Figure 2 Irregular interval-valued fuzzy graph

Example 27 Consider a graph 119866lowast such that

119881 = V1 V2 V3 119864 = V

1V2 V2V3 V1V3 (23)

Let 119860 be an interval-valued fuzzy subset of119881 and let 119861 be aninterval-valued fuzzy subset of 119864 sube 119881 times 119881 defined by

[[[

[

V1

V2

V3

120583minus119860

02 02 03

120583+119860

06 07 04

]]]

]

[[[

[

V1V2V1V3V2V3

120583minus119861

01 01 02

120583+119861

02 02 03

]]]

]

(24)

By routine computations we have deg(V1) = [05 11]

deg(V2) = [05 10] and deg(V

3) = [04 13] It is clear that

119866 as shown in Figure 2 is an irregular interval-valued fuzzygraph

Definition 28 Let 119866 be an interval-valued fuzzy graph Ifthere is a vertex which is adjacent to vertices with distinctclosed neighbourhood degrees then 119866 is called a totallyirregular interval-valued fuzzy graph

Example 29 Consider an interval-valued fuzzy graph119866 suchthat

119881 = V1 V2 V3 V4 V5

119864 = V1V2 V2V3 V2V4 V3V1 V3V4 V4V1 V4V5

(25)

By routine computations we have deg[V1] = [14 24]

deg[V2] = [14 24] deg[V

3] = [14 24] deg[V

4] = [16 26]

and deg[V5] = [06 08] It is clear from calculations that 119866 as

1 2

34

[04 06]

[04 06]

[02 02]

[03 05]

[03 07]

[01 02]

[02 02][01 04]

[0204]

[01 03]

[02 03]

[03 05]

5

Figure 3 119866 is totally irregular

1 2

4 3

[02 06]

[05 05]

[03 07]

[01 04] [01 04]

[04 04]

[01 02]

[01 02]

Figure 4 119866 is neighbourly irregular

shown in Figure 3 is a totally irregular interval-valued fuzzygraph

Definition 30 A connected interval-valued fuzzy graph 119866 issaid to be neighbourly irregular if every two adjacent verticesof 119866 have distinct open neighbourhood degree

Example 31 Consider an interval-valued fuzzy graph 119866 suchthat

119881 = V1 V2 V3 V4

119864 = V1V2 V2V3 V3V4 V4V1

(26)

By routine computations we have deg(V1) = [08 12]

deg(V2) = [06 10] deg(V

3) = [08 12] and deg(V

4) =

[06 10] Hence 119866 as shown in Figure 4 is neighbourlyirregular

Definition 32 A connected interval-valued fuzzy graph 119866 issaid to be neighbourly totally irregular if every two adjacentvertices of 119866 have distinct closed neighbourhood degree

Journal of Applied Mathematics 7

1 2

4 3

[03 06] [04 05]

[04 05] [02 07]

[01 02]

[01 02]

[01 04] [01 04]

Figure 5 119866 is neighbourly totally irregular

Example 33 Consider an interval-valued fuzzy graph119866 suchthat

119881 = V1 V2 V3 V4

119864 = V1V2 V2V3 V3V4 V4V1

(27)

By routine computations we have deg[V1] = [11 16]

deg[V2] = [09 18] deg[V

3] = [10 17] and deg[V

4] =

[09 18] Hence119866 as shown in Figure 5 is neighbourly totallyirregular

Definition 34 Let 119866 be a connected interval-valued fuzzygraph 119866 is called highly irregular if every vertex of 119866 isadjacent to vertices with distinct neighbourhood degrees

Example 35 Consider an interval-valued fuzzy graph119866 suchthat

119881 = V1 V2 V3 V4 V5 V6

119864 = V1V2 V2V3 V2V6 V3V4 V3V5 V4V5 V5V1

(28)

By routine computations we have deg(V1) = [04 08]

deg(V2) = [06 17] deg(V

3) = [09 13] deg(V

4) = [06 11]

deg(V5) = [10 18] and deg(V

6) = [01 04] Clearly 119866 as

shown in Figure 6 is highly irregular

Example 36 Consider an interval-valued fuzzy graph119866 suchthat

119881 = V1 V2 V3 V4

119864 = V1V2 V2V3 V3V4 V4V1

(29)

By routine computations we have deg(V1) = [06 10]

deg(V2) = [08 09] deg(V

3) = [06 10] and deg(V

4) =

1 2

4

35

[02 02]

[01 03]

[03 04]

[02

04

][02

04]

[05 05]

[02 06] [01 04]

[01 04]

[03 07]

[01 0

4]

[02 02]

[02 03]

6

Figure 6 119866 is highly irregular

[08 09]We see that every two adjacent vertices have distinctopen neighbourhood degree But the vertex V

2adjacent to

the vertices V1and V

3has the same neighbourhood degree

that is deg(V1) = deg(V

3) Hence 119866 as shown in Figure 7 is

neighbourly irregular but not highly irregular

Remark 37 A neighbourly irregular interval-valued fuzzygraph may not be highly irregular

Theorem38 An interval-valued fuzzy graph119866 is highly irreg-ular and neighbourly irregular interval-valued fuzzy graph ifand only if the neighbourhood degrees of all the vertices of 119866are distinct

Proof Let 119866 be an interval-valued fuzzy graph with 119899-vertices V

1 V2 V

119899 Assume that 119866 is highly irregular and

neighbourly irregular

Claim 1 The neighbourhood degrees of all vertices of 119866

are distinct Let deg(V119894) = [119896

119894 119897119894] 119894 = 1 2 119899 Let

the adjacent vertices of V1be V2 V3 V

119899with neighbour-

hood degrees [1198962 1198972] [1198963 1198973] [119896

119899 119897119899] respectively Then

1198962

= 1198963

= sdot sdot sdot = 119896119899

and 1198972

= 1198973

= sdot sdot sdot = 119897119899 since 119866 is highly

irregular Also 1198961

= 1198962

= 1198963

= sdot sdot sdot = 119896119899and 1198971

= 1198972

= 1198973

= sdot sdot sdot = 119897119899

since 119866 is neighbourly irregular Hence the neighbourhooddegree of all the vertices of 119866 is distinct

8 Journal of Applied Mathematics

1 2

34

[03 04]

[01 04]

[04 05]

[01 02]

[01 02]

[02 05]

[05 05]

[01 04]

Figure 7 119866 is neighbourly irregular but not highly irregular

Conversely assume that the neighbourhood degrees of allthe vertices of 119866 are distinct

Claim 2 119866 is highly irregular and neighbourly irregularinterval-valued fuzzy graph

Let deg(V119894) = [119896

119894 119897119894] 119894 = 1 2 119899 Given that

1198961

= 1198962

= 1198963

= sdot sdot sdot = 119896119899and 1198971

= 1198972

= 1198973

= sdot sdot sdot = 119897119899 which implies

that every two adjacent vertices have distinct neighbourhooddegrees and to every vertex the adjacent vertices have distinctneighbourhood degrees

Theorem 39 An interval-valued fuzzy graph 119866 of 119866lowast where119866lowast is a cycle with 3 vertices that is neighbourly irregular andhighly irregular if and only if the lower and upper membershipvalues of the vertices between every pair of vertices are alldistinct

Proof Assume that lower and upper membership values ofthe vertices are all distinct

Claim 1 119866 is neighbourly irregular and highly irregularinterval-valued fuzzy graph

Let V119894 V119895 V119896

isin 119881 Given that 120583minus119860(V119894) = 120583minus119860(V119895) =

120583minus

119860(V119896) and 120583

+

119860(V119894) = 120583+

119860(V119895) = 120583+

119860(V119896) which implies that

sum119909isin119873(119909)

120583minus119860(V119894) = sum119909isin119873(119909)

120583minus119860(V119895) = sum119909isin119873(119909)

120583minus119860(V119896) and

sum119909isin119873(119909)

120583+119860(V119894) = sum119909isin119873(119909)

120583+119860(V119895) = sum119909isin119873(119909)

120583+119860(V119896) That is

deg(V119894) = deg(V

119895) = deg(V

119896) Hence 119866 is neighbourly

irregular and highly irregularConversely assume that 119866 is neighbourly irregular and

highly irregular

Claim 2 Lower and upper membership values of the verticesare all distinct

Let deg(V119894) = [119896

119894 119897119894] 119894 = 1 2 119899 Suppose that lower

and upper membership value of any two vertices are the

1

23

[04 06]

[02 06]

[02 07] [02 06]

[04 06]

[04 06]

Figure 8 119866 is not neighbourly irregular but it is complete

same Let V1 V2

isin 119881 Let 120583minus119860(V1) = 120583minus

119860(V2) and 120583+

119860(V1) =

120583+119860(V2) Then deg(V

1) = deg(V

2) since 119866lowast is cycle which is

a contradiction to the fact that119866 is neighbourly irregular andhighly irregular interval-valued fuzzy graph Hence lowermembership and upper membership value of the vertices areall distinct

Remark 40 A complete interval-valued fuzzy graph may notbe neighbourly irregular

Example 41 Consider an interval-valued fuzzy graph 119866 suchthat

119881 = V1 V2 V3 119864 = V

1V2 V2V3 V1V3 (30)

By routine computations we have deg(V1) = [06 13]

deg(V2) = [06 13] and deg(V

3) = [08 12] We see that

neighbourhood degree of V1and V

2is not distinct Hence

119866 as shown in Figure 8 is not neighbourly irregular but it iscomplete

Remark 42 A neighbourly total irregular interval-valuedfuzzy graph may not be neighbourly irregular

Example 43 Consider an interval-valued fuzzy graph119866 suchthat

119881 = V1 V2 V3 V4 119864 = V

1V2 V2V3 V3V4 V4V1 (31)

By routine computations we have deg(V1) = [07 09]

deg(V2) = [08 11] deg(V

3) = [07 09] deg(V

4) = [08 11]

deg[V1] = [11 15] deg[V

2] = [11 15] deg[V

3] = [11 14]

and deg[V4] = [12 16] We see that deg[V

1] = deg[V

2]

Hence 119866 as shown in Figure 9 is neighbourly irregular butnot a neighbourly total irregular

Journal of Applied Mathematics 9

1 2

34

[04 06]

[01 04]

[04 05] [01 02]

[01 02]

[04 05]

[03 04]

[01 04]

Figure 9 119866 is neighbourly irregular but not neighbourly totalirregular

Proposition 44 If an interval-valued fuzzy graph 119866 is neigh-bourly irregular and [120583minus

119860 120583+119860] is a constant function then it is

neighbourly totally irregular

Proof Assume that 119866 is a neighbourly irregular interval-valued fuzzy graph Then the open neighbourhood degreesof every two adjacent vertices are distinct Let V

119894 V119895

isin 119881 beadjacent vertices with distinct open neighbourhood degrees[1198961 1198971] and [119896

2 1198972] where 119896

1= 1198962 1198971

= 1198972 Let us assume that

(1205831(V119894) ]1(V119894)) = (120583

1(V119895) ]1(V119895)) = [119888

1 1198882] where 119888

1 1198882are

constant and 1198881 1198882isin [0 1] Therefore deg

120583minus

[V119894] = deg

120583minus

(V119894) +

1205831(V119894) = 119896

1+ 1198881 deg120583+

[V119894] = deg

120583+

(V119894) + ]1(V119894) = 119897

1+

1198882deg120583minus

[V119895] = deg

120583minus

(V119895) + 1205831(V119895) = 1198962+ 1198881 and deg

120583+

[V119895] =

deg120583+

(V119895) + ]1(V119895) = 1198972+ 1198882

Claim Consider that deg120583minus

[V119894] = deg

120583minus

[V119895] and deg

120583+

[V119894] =

deg120583+

[V119895] Suppose that deg

120583minus

[V119894] = deg

120583minus

[V119895] and

deg120583+

[V119894] = deg

120583+

[V119895] Consider that

deg120583minus

[V119894] = deg

120583minus

[V119895]

1198961+ 1198881= 1198962+ 1198881

1198961minus 1198962= 1198881minus 1198881= 0

1198961= 1198962 which is a contradiction to 119896

1= 1198962

(32)

Therefore deg120583minus

[V119894] = deg

120583minus

[V119895] Similarly we consider that

deg120583+

[V119894] = deg

120583+

[V119895]

1198971+ 1198882= 1198972+ 1198882

1198971minus 1198972= 1198882minus 1198882= 0

1198971= 1198972 which is a contradiction to 119897

1= 1198972

(33)

Therefore deg120583+

[V119894] = deg

120583+

[V119895] Hence 119866 is a neighbourly

totally irregular interval-valued fuzzy graph

Theorem 45 If an interval-valued fuzzy graph 119866 is neigh-bourly totally irregular and [120583minus

119860 120583+119860] is a constant function

then it is a neighbourly irregular interval-valued fuzzy graph

Proof Assume that119866 is a neighbourly total irregular interval-valued fuzzy graph Then the closed neighbourhood degreeof every two adjacent vertices is distinct Let V

119894 V119895

isin 119881

and deg[V119894] = [119896

1 1198971] deg[V

119895] = [119896

2 1198972] where 119896

1= 1198962

and 1198971

= 1198972 Assume that (120583

1(V119894) ]1(V119894)) = [119888

1 1198882] and

(1205831(V119895) ]1(V119895)) = [119888

1 1198882] where 119888

1 1198882

isin [0 1] are constantand deg[V

119894] = deg[V

119895]

Claim Consider that deg(V119894) = deg(V

119895)

Given that deg[V119894] = deg[V

119895] which implies deg

120583minus

[V119894] =

deg120583minus

[V119895] and deg

120583+

[V119894] = deg

120583+

[V119895] now we consider that

deg120583minus

[V119894] = deg

120583minus

[V119895]

1198961+ 1198881

= 1198962+ 1198881

1198961

= 1198962

(34)

We now consider that

deg120583+

[V119894] = deg

120583+

[V119895]

1198971+ 1198882

= 1198972+ 1198882

1198971

= 1198972

(35)

That is the neighbourhood degrees of adjacent vertices of 119866are distinct Hence neighbourhood degree of every pair ofadjacent vertices is distinct in 119866

Proposition 46 If an interval-valued fuzzy graph 119866 is neigh-bourly irregular and neighbourly totally irregular then [120583

minus

119860 120583+

119860]

need not be a constant function

Remark 47 If 119866 is a neighbourly irregular interval-valuedfuzzy graph then interval-valued subgraph 119867 = (119860

1015840 1198611015840) of119866 may not be neighbourly irregular

Remark 48 If 119866 is a totally irregular interval-valued fuzzygraph then interval-valued fuzzy subgraph 119867 = (1198601015840 1198611015840) of119866 may not be totally irregular

5 Relationship between IFGs and IVFGs

In 2003 Deschrijver and Kerre [31] established the rela-tionships between some extensions of fuzzy sets In thissection we present the relationship between extensions offuzzy graphs Shannon and Atanassov [32] introduced thenotion of an intuitionistic fuzzy graph Some operations onintuitionistic fuzzy graphs are discussed in [33]

Definition 49 (see [32]) By an intuitionistic fuzzy graph (IFGfor short) 119866 of a graph 119866

lowast we mean a pair 119866 = (119860 119861)

10 Journal of Applied Mathematics

Fuzzy sets

Interval-valued

fuzzy sets

Intuitionisticfuzzy sets

Interval-valued

fuzzy graphs

Intuitionisticfuzzy graphs

Figure 10 Links between models

Interval-valued

fuzzy graphs

Intuitionistic

fuzzy graphs

Fuzzy graphs

Figure 11 Link between IFGs and IVFGs

where 119860 = (120583119860 ]119860) is an intuitionistic fuzzy set on 119881 and

119861 = (120583119861 ]119861) is an intuitionistic fuzzy relation on 119864 such that

120583119861(119909119910) le min (120583

119860(119909) 120583

119860(119910))

]119861(119909119910) le max (]

119860(119909) ]

119860(119910))

(36)

for all 119909119910 isin 119864 The class of all IFGs on 119866lowast will be denoted byIFG(119866lowast)

Theorem 50 (see [33]) If 1198661and 119866

2are intuitionistic fuzzy

graphs then 1198661cap 1198662 1198661cup 1198662 and 119866

1are intuitionistic fuzzy

graphs

Ju and Wang introduced the notion of interval-valuedfuzzy graph (IVFG for short) in [15] Some operations oninterval-valued fuzzy graphs are discussed in [18] The classof all IVFGs on 119866

lowast will be denoted byIVFG(119866lowast)

Theorem 51 (see [19]) If 1198661and 119866

2are interval-valued fuzzy

graphs then1198661cap11986621198661cup1198662 and119866

1are interval-valued fuzzy

graphs

Lemma 52 (IVFG(119866lowast) cup cap) and (IFG(119866lowast) cup cap) arecomplete lattices

Theorem 53 The mapping

1205951 IVFG (119866

lowast) 997888rarr IFG (119866

lowast) (37)

defined by

1205951(119861) = ((119909 120583

minus

119860(119909) 1 minus 120583

+

119860(119909)) | 119909 isin 119881

(119909119910 120583minus

119861(119909119910) 1 minus 120583

+

119861(119909119910)) | 119909119910 isin 119864)

(38)

where119861 = ((119909 120583

minus

119860(119909) 120583

+

119860(119909)) | 119909 isin 119881

(119909119910 120583minus

119861(119909119910) 120583

+

119861(119909119910)) | 119909119910 isin 119864)

(39)

is an isomorphism between lattices (IVFG(119866lowast) cup cap) and(IFG(119866lowast) cup cap)

Remark 54 From a pure mathematical point of viewTheorem 53 shows that the two concepts intuitionistic fuzzygraphs and interval-valued fuzzy graphs are equivalent

In Figures 10 and 11 we present the relationships that existbetween different models In these figures a double arrowbetween two theories means that they are equivalent a singlearrow 119883 rarr 119884 denotes that 119884 is an extension of 119883 InFigure 10 a dash arrow 119883[119884 denotes that model 119884 is basedon the previous model 119883

6 Conclusions

Graph theory has several interesting applications in systemanalysis operations research computer applications andeconomics Since most of the time the aspects of graphproblems are uncertain it is nice to deal with these aspectsvia the methods of fuzzy systems It is known that fuzzygraph theory has numerous applications in modern scienceand engineering especially in the field of information theoryneural networks expert systems cluster analysis medicaldiagnosis traffic engineering network routing town plan-ning and control theory Since interval-valued fuzzy settheory is an increasingly popular extension of fuzzy settheory where traditional [0 1]-valued membership degreesare replaced by intervals in [0 1] that approximate the(unknown) membership degrees specific types of interval-valued fuzzy graphs have been introduced and investigatedin this paper The natural extension of this research work isthe application of interval-valued fuzzy graphs in the area ofsoft computing including neural networks expert systemsdatabase theory and geographical information systems

Acknowledgments

The authors are thankful to the referees for their valuablecomments The authors are also thankful to Professor SyedMansoor Sarwar and Dr Faisal Aslam for their valuablesuggestions

References

[1] S G Shirinivas S Vetrivel and N M Elango ldquoApplications ofgraph theory in computer sciencemdashan overviewrdquo InternationalJournal of Engineering Science and Technology vol 2 no 9 pp4610ndash4621 2010

Journal of Applied Mathematics 11

[2] N Deo Graph Theory with Applications to Engineering andComputer Science Prentice Hall Englewood Cliffs NJ USA1990

[3] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoningmdashIrdquo Information Sciencesvol 8 pp 199ndash249 1975

[4] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[5] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 pp338ndash353 1965

[6] J M Mendel Uncertain Rule-Based Fuzzy Logic Systems Intro-duction and New Directions Prentice-Hall Upper Saddle RiverNJ USA 2001

[7] M B Gorzałczany ldquoA method of inference in approximatereasoning based on interval-valued fuzzy setsrdquo Fuzzy Sets andSystems vol 21 no 1 pp 1ndash17 1987

[8] M B Gorzałczany ldquoAn interval-valued fuzzy inferencemethodmdashsome basic propertiesrdquo Fuzzy Sets and Systems vol31 no 2 pp 243ndash251 1989

[9] M K Roy and R Biswas ldquoI-V fuzzy relations and Sanchezrsquosapproach for medical diagnosisrdquo Fuzzy Sets and Systems vol47 no 1 pp 35ndash38 1992

[10] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986

[11] A Kauffman Introduction a la Theorie des Sous-emsemblesFlous vol 1 Masson et Cie 1973

[12] A Rosenfeld ldquoFuzzy graphsrdquo in Fuzzy Sets and Their Applica-tions L A Zadeh K S Fu and M Shimura Eds pp 77ndash95Academic Press New York NY USA 1975

[13] J N Mordeson and C-S Peng ldquoOperations on fuzzy graphsrdquoInformation Sciences vol 79 no 3-4 pp 159ndash170 1994

[14] M S Sunitha and A V Kumar ldquoComplement of a fuzzy graphrdquoIndian Journal of Pure and Applied Mathematics vol 33 no 9pp 1451ndash1464 2002

[15] H Ju and L Wang ldquoInterval-valued fuzzy subsemigroupsand subgroups associated by intervalvalued fuzzy graphsrdquo inProceedings of the WRI Global Congress on Intelligent Systems(GCIS rsquo09) pp 484ndash487 Xiamen China May 2009

[16] MAkram ldquoBipolar fuzzy graphsrdquo Information Sciences vol 181no 24 pp 5548ndash5564 2011

[17] M Akram ldquoBipolar fuzzy graphs with applicationsrdquo KnowledgeBased Systems vol 39 pp 1ndash8 2013

[18] M Akram ldquoInterval-valued fuzzy line graphsrdquoNeural Comput-ing and Applications vol 21 pp 145ndash150 2012

[19] M Akram and W A Dudek ldquoInterval-valued fuzzy graphsrdquoComputers amp Mathematics with Applications vol 61 no 2 pp289ndash299 2011

[20] M Akram and B Davvaz ldquoStrong intuitionistic fuzzy graphsrdquoFilomat vol 26 no 1 pp 177ndash196 2012

[21] M Akram K H Dar and K P Shum ldquoInterval-valued (120572120573)-fuzzy K-algebrasrdquo Applied Soft Computing Journal vol 11 no 1pp 1213ndash1222 2011

[22] T AL-Hawary ldquoComplete fuzzy graphsrdquo International Journalof Mathematical Combinatorics vol 4 pp 26ndash34 2011

[23] P Bhattacharya ldquoSome remarks on fuzzy graphsrdquo PatternRecognition Letters vol 6 no 5 pp 297ndash302 1987

[24] S M Chen ldquoInterval-valued fuzzy hypergraph and fuzzypartitionrdquo IEEE Transactions on Systems Man and CyberneticsB vol 27 no 4 pp 725ndash733 1997

[25] G Deschrijver and C Cornelis ldquoRepresentability in interval-valued fuzzy set theoryrdquo International Journal of UncertaintyFuzziness and Knowledge-Based Systems vol 15 no 3 pp 345ndash361 2007

[26] A N Gani and S R Latha ldquoOn irregular fuzzy graphsrdquoAppliedMathematical Sciences vol 6 no 9ndash12 pp 517ndash523 2012

[27] X Ma J Zhan B Davvaz and Y B Jun ldquoSome kinds of (isin isinor119902)-interval-valued fuzzy ideals of BCI-algebrasrdquo InformationSciences vol 178 no 19 pp 3738ndash3754 2008

[28] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Studies in Fuzziness and SoftComputing PhysicaHeidelberg Germany 1998

[29] F Riaz and KM Ali ldquoApplications of graph theory in computersciencerdquo in Proceedings of the 3rd International Conferenceon Computational Intelligence Communication Systems andNetworks (CICSyN rsquo11) pp 142ndash145 Bali Indonesia July 2011

[30] M Behzad and G Chartrand ldquoNo graph is perfectrdquo TheAmerican Mathematical Monthly vol 74 pp 962ndash963 1967

[31] G Deschrijver and E E Kerre ldquoOn the relationship betweensome extensions of fuzzy set theoryrdquo Fuzzy Sets and Systemsvol 133 no 2 pp 227ndash235 2003

[32] A Shannon and K T Atanassov ldquoA first step to a theory ofthe intuitionistic fuzzy graphsrdquo in Proceeding of the FUBESTD Lakov Ed pp 59ndash61 Sofia Bulgaria 1994

[33] R Parvathi M G Karunambigai and K T Atanassov ldquoOper-ations on intuitionistic fuzzy graphsrdquo in Proceedings of theIEEE International Conference on Fuzzy Systems pp 1396ndash1401August 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

Interval-valued fuzzy relations reflect the idea that mem-bership grades are often not precise and the intervals repre-sent such uncertainty

Definition 3 (see [15]) By an interval-valued fuzzy graph119866 ofa graph 119866

lowast we mean a pair 119866 = (119860 119861) where 119860 = [120583minus119860 120583+119860]

is an interval-valued fuzzy set on 119881 and 119861 = [120583minus119861 120583+119861] is an

interval-valued fuzzy relation on 119864 such that

120583minus

119861(119909119910) le min (120583

minus

119860(119909) 120583

minus

119860(119910))

120583+

119861(119909119910) le min (120583

+

119860(119909) 120583

+

119860(119910))

(1)

for all 119909119910 isin 119864

Throughout this paper 119866lowast is a crisp graph and 119866 is aninterval-valued fuzzy graph

3 Balanced Interval-Valued Fuzzy Graphs

Definition 4 Let 119866 be an interval-valued fuzzy graph Theneighbourhood degree of a vertex 119909 in119866 is defined by deg(119909) =

[deg120583minus

(119909) deg120583+

(119909)] where deg120583minus

(119909) = sum119910isin119873(119909)

120583minus119860(119910) and

deg120583+

(119909) = sum119910isin119873(119909)

120583+119860(119910) Notice that 120583minus

119861(119909119910) gt 0 120583+

119861(119909119910) gt

0 for 119909119910 isin 119864 and 120583minus119861(119909119910) = 120583+

119861(119909119910) = 0 for 119909119910 notin 119864

Definition 5 Let 119866 = (119860 119861) be an interval-valued fuzzygraph on119866lowast If all the vertices have the same open neighbour-hood degree 119899 then 119866 is called an 119899-regular interval-valuedfuzzy graph The open neighbourhood degree of a vertex 119909

in 119866 is defined by deg(119909) = [deg120583minus

(119909) deg120583+

(119909)] wheredeg120583minus

(119909) = sum119910isin119873(119909)

120583minus119860(119910) and deg

120583+

(119909) = sum119910isin119873(119909)

120583+119860(119910)

Definition 6 Let 119866 be an interval-valued fuzzy graph Theclosed neighbourhood degree of a vertex 119909 is defined bydeg[119909] = [deg

120583minus

[119909] deg120583+

[119909]] where

deg120583minus[119909] = deg

120583minus(119909) + 120583

minus

119860(119909)

deg120583+[119909] = deg

120583+(119909) + 120583

+

119860(119909)

(2)

If all the vertices have the same closed neighbourhood degree119898 then 119866 is called a 119898-totally regular interval-valued fuzzygraph

Example 7 Consider a graph 119866lowast such that 119881 = 119886 119887 119888 119889119864 = 119886119887 119887119888 119888119889 119886119889 Let 119860 be an interval-valued fuzzy subsetof 119881 and let119861 be an interval-valued fuzzy subset of119864 sube 119881times119881

defined by

[[[

[

119886 119887 119888 119889

120583minus119860

03 03 03 03

120583+119860

05 05 05 05

]]]

]

d c

ba

[01 02]

[01 04][01 04]

[01 02]

[03 05] [03 05]

[03 05][03 05]

G

Figure 1 119866 is regular and totally regular

[[[

[

119886119887 119887119888 119888119889 119889119886

120583minus119861

01 01 01 01

120583+119861

02 04 02 04

]]]

]

(3)

Routine computations show that an interval-valued fuzzygraph 119866 as shown in Figure 1 is both regular and totallyregular

Example 8 Consider a graph 119866lowast such that 119881 = V

1 V2 V3

119864 = V1V2 V1V3 Let 119860 be an interval-valued fuzzy subset of

119881 and let 119861 be an interval-valued fuzzy subset of 119864 defined by

120583minus

119860(V1) = 04 120583

minus

119860(V2) = 07 120583

minus

119860(V3) = 06

120583+

119860(V1) = 04 120583

+

119860(V2) = 08 120583

+

119860(V3) = 07

120583minus

119861(V1V2) = 02 120583

minus

119861(V1V3) = 02 120583

+

119861(V1V2) = 03

120583+

119861(V1V3) = 04

(4)

Routine computations show that an interval-valued fuzzygraph 119866 is neither totally regular nor regular

Definition 9 We define the order 119874(119866) and size 119878(119866) of aninterval-valued fuzzy graph 119866 = (119860 119861) by

119874 (119866) = sum119909isin119881

1 + 120583+119860(119909) minus 120583minus

119860(119909)

2

119878 (119866) = sum119909119910isin119864

1 + 120583+119861(119909119910) minus 120583minus

119861(119909119910)

2

(5)

4 Journal of Applied Mathematics

Definition 10 An interval-valued fuzzy graph 119866 = (119860 119861) iscalled complete if

120583minus

119861(119909119910) = min (120583

minus

119860(119909) 120583

minus

119860(119910))

120583+

119861(119909119910) = max (120583

+

119860(119909) 120583

+

119860(119910)) forall119909 119910 isin 119881

(6)

Example 11 Consider a graph119866lowast such that119881 = 119909 119910 119911 119864 =

119909119910 119910119911 119911119909 Let119860 be an interval-valued fuzzy subset of119881 andlet 119861 be an interval-valued fuzzy subset of 119864 defined by

120583minus

119860(119909) = 03 120583

minus

119860(119910) = 04 120583

minus

119860(119911) = 05

120583+

119860(119909) = 05 120583

+

119860(119910) = 07 120583

+

119860(119911) = 06

120583minus

119861(119909119910) = 03 120583

minus

119861(119910119911) = 04 120583

minus

119861(119911119909) = 03

120583+

119861(119909119910) = 05 120583

+

119861(119910119911) = 06 120583

+

119861(119911119909) = 05

(7)

Routine computations show that 119866 is both completeand totally regular interval-valued fuzzy graph but 119866 is notregular since deg(119909) = deg(119911) = deg(119910)

Theorem 12 Every complete interval-valued fuzzy graph istotally regular

Theorem 13 Let119866 = (119860 119861) be an interval-valued fuzzy graphof a graph 119866

lowast Then119860 = [120583minus119860 120583+119860] is a constant function if and

only if the following statements are equivalent

(a) 119866 is a regular interval-valued fuzzy graph(b) 119866 is a totally regular interval-valued fuzzy graph

Proof Suppose that 119860 = [120583minus119860 120583+119860] is a constant function Let

120583minus119860(119909) = 119888

1and 120583+

119860(119909) = 119888

2for all 119909 isin 119881

(a) rArr (b) Assume that 119866 is an 119899-regular interval-valuedfuzzy graph Then deg

120583minus

(119909) = 1198991and deg

120583+

(119909) = 1198992for all

119909 isin 119881 So

deg120583minus[119909] = deg

120583minus(119909) + 120583

minus

119860(119909)

deg120583+[119909] = deg

120583+(119909) + 120583

+

119860(119909) forall119909 isin 119881

(8)

Thus

deg120583minus[119909] = 119899

1+ 1198881 deg

120583+[119909] = 119899

2+ 1198882 forall119909 isin 119881 (9)

Hence 119866 is a totally regular interval-valued fuzzy graph(b) rArr (a) Suppose that 119866 is a totally regular interval-

valued fuzzy graph Then

deg120583minus[119909] = 119896

1 deg

120583+[119909] = 119896

2 forall119909 isin 119881 (10)

or

deg120583minus(119909) + 120583

minus

119860(119909) = 119896

1

deg120583+(119909) + 120583

+

119860(119909) = 119896

2 forall119909 isin 119881

(11)

or

deg120583minus(119909) + 119888

1= 1198961 deg

120583+(119909) + 119888

2= 1198962 forall119909 isin 119881 (12)

or

deg120583minus(119909) = 119896

1minus 1198881 deg

120583+(119909) = 119896

2minus 1198882 forall119909 isin 119881 (13)

Thus 119866 is a regular interval-valued fuzzy graph Hence (a)and (b) are equivalent

The converse part is obvious

Theorem 14 Let 119866 be an interval-valued fuzzy graph where acrisp graph 119866lowast is an odd cycle Then 119866 is a regular interval-valued fuzzy graph if and only if 119861 is a constant function

Proof If 119861 = [120583minus119861 120583+119861] is a constant function say 120583minus

119861= 1198881and

120583+119861

= 1198882for all 119909119910 isin 119864 then deg

120583minus

(119909) = 21198881and deg

120583+

(119909) =

21198882for every119909 isin 119881 Hence119866 is a regular interval-valued fuzzy

graphConversely suppose that 119866 is a (119896

1 1198962)-regular interval-

valued fuzzy graph Let 1198901 1198902 119890

2119899+1be the edges of 119866 in

that order Let 120583minus119861(1198901) = 1198881 120583minus119861(1198902) = 119896

1minus 1198881 120583minus119861(1198903) = 119896

1minus

(1198961minus 1198881) = 1198881 120583minus119861(1198904) = 1198961minus 1198881 and so on Therefore

120583minus

119861(119890119894) =

1198881 if 119894 is odd

1198961minus 1198881 if 119894 is even

(14)

Thus 120583minus119861(1198901) = 120583minus119861(1198902119899+1

) = 1198881 So if 119890

1and 1198902119899+1

incident at avertex V

1 then deg

120583minus

(V1) = 1198961 deg120583minus

(1198901) + deg

120583minus

(1198902119899+1

) = 1198961

1198881+ 1198881= 1198961 21198881= 1198961 and 119888

1= 11989612 This shows that 120583minus

119861is a

regular functionSimilarly let 120583+

119861(1198901) = 1198882 120583+119861(1198902) = 1198962minus 1198882 120583+119861(1198903) = 1198962minus

(1198962minus 1198882) = 1198882 120583+119861(1198904) = 1198962minus 1198882 and so on Therefore

120583+

119861(119890119894) =

1198882 if 119894 is odd

1198962minus 1198882 if 119894 is even

(15)

Thus 120583+119861(1198902) = 120583+

119861(1198902119899+1

) = 1198882 So if 119890

2and 1198902119899

incident ata vertex V

2 then deg

120583+

(V2) = 1198962 deg120583+

(1198902) + deg

120583+

(1198902119899+1

) =

1198962 1198882+ 1198882

= 1198962 21198882

= 1198962 and 119888

2= 11989622 This shows that

120583+119861is a constant function Hence 119861 = [120583minus

119861 120583+119861] is a constant

function

We state the following characterization without its proof

Theorem 15 Let 119866 be an interval-valued fuzzy graph where acrisp graph 119866lowast is an even cycle Then 119866 is a regular interval-valued fuzzy graph if and only if either 119861 = [120583minus

119861 120583+119861] is a

constant function or alternate edges have the samemembershipvalues

Definition 16 The density of an interval-valued fuzzygraphs 119866 is 119863(119866) = (119863

minus(119866) 119863+(119866)) where 119863minus(119866) =(2sum119909119910isin119881

(120583minus119861(119909119910)))(sum

119909119910isin119881(120583minus119860(119909) and 120583minus

119860(119910))) for 119909 119910 isin 119881

and119863+(119866) = (2sum119909119910isin119881

(120583+119861(119909119910)))(sum

119909119910isin119881(120583+119860(119909)and120583+

119860(119910))) for

119909 119910 isin 119881 An interval-valued fuzzy graph 119866 is balanced if119863(119867) le 119863(119866) that is 119863minus(119867) le 119863minus(119866) 119863+(119867) le 119863+(119866)

for all subgraphs119867 of 119866 An interval-valued fuzzy graph 119866 isstrictly balanced if for every 119909 119910 isin 119881 119863(119867) = 119863(119866) for allnonempty subgraphs

Journal of Applied Mathematics 5

Example 17 Consider the regular interval-valued fuzzy graph119866which is given in Example 7 Routine calculations show that119863minus(119866) = 067 and 119863+(119866) = 12 Thus 119863(119866) = (067 12)

Consider that 1198671

= 119886 119887 119888 1198672

= 119886 119888 1198673

= 119886 119889 and1198674

= 119887 119888 are a nonempty subgraphs of 119866 Then 119863(1198671) =

(067 12) 119863(1198672) = (0 0) 119863(119867

3) = (067 16) and 119863(119867

4) =

(067 16) It is easy to see that regular interval-valued fuzzygraph is not balanced

Remark 18 Every regular interval-valued fuzzy graph maynot be balanced

Example 19 Consider the regular interval-valued fuzzygraph 119866 which is given in Example 11 Routine calculationsshow that 119863minus(119866) = 2 and 119863+(119866) = 2 Thus 119863(119866) = (2 2)Consider119867

1= 119909 119910119867

2= 119909 119911119867

3= 119910 119911 be a nonempty

subgraphs of 119866 Then 119863(1198671) = (2 2) 119863(119867

2) = (2 2) and

119863(1198673) = (2 2) It is easy to see that complete interval-valued

fuzzy graph is balanced 119866 is also strictly balanced

Proposition 20 Any complete interval-valued fuzzy graph isbalanced

Proposition 21 Let 119866 be a self-complementary interval-valued fuzzy graph Then 119863(119866) = (1 1)

Proposition 22 Let 1198661and 119866

2be two balanced interval-

valued fuzzy graphs Then 1198661times 1198662is balanced if and only if

119863(1198661) = 119863(119866

2) = 119863(119866

1times 1198662)

Theorem23 Let119866 be a strictly balanced interval-valued fuzzygraph and let119866 be its complement then119863(119866)+119863(119866) = (2 2)

Proof Let 119866 be a strictly balanced interval-valued fuzzygraph and 119866 its complement Let119867 be a nonempty subgraphof 119866 Since 119866 is strictly balanced 119863(119866) = 119863(119867) for every119867 sube 119866 and 119909 119910 isin 119881 In 119866

120583minus119861(119909119910) = 120583

minus

119860(119909) and 120583

minus

119860(119910) minus 120583

minus

119861(119909119910) (16)

120583+119861(119909119910) = 120583

+

119860(119909) and 120583

+

119860(119910) minus 120583

+

119861(119909119910) (17)

for every 119909 119910 isin 119881 Dividing (16) by 120583minus119860(119909) and 120583minus

119860(119910) we get

120583minus119861(119909119910)

120583minus119860(119909) and 120583minus

119860(119910)

= 1 minus120583minus119861(119909119910)

120583minus119860(119909) and 120583minus

119860(119910)

for every 119909 119910isin119881

(18)

and dividing (17) by 120583+119860(119909) and 120583+

119860(119910) we get

120583+119861(119909119910)

120583+119860(119909) and 120583+

119860(119910)

= 1 minus120583+119861(119909119910)

120583+119860(119909) and 120583+

119860(119910)

for every 119909 119910isin119881

(19)

Then

sum119909119910isin119881

120583minus119861(119909119910)

120583minus119860(119909) and 120583minus

119860(119910)

= 1 minus sum119909119910isin119881

120583minus119861(119909119910)

120583minus119860(119909) and 120583minus

119860(119910)

for every 119909 119910 isin 119881

sum119909119910isin119881

120583+119861(119909119910)

120583+119860(119909) and 120583+

119860(119910)

= 1 minus sum119909119910isin119881

120583+119861(119909119910)

120583+119860(119909) and 120583+

119860(119910)

for every 119909 119910 isin 119881

(20)

Multiplying both sides the above equations by 2

2 sum119909119910isin119881

120583minus119861(119909119910)

120583minus119860(119909) and 120583minus

119860(119910)

= 2 minus 2 sum119909119910isin119881

120583minus119861(119909119910)

120583minus119860(119909) and 120583minus

119860(119910)

for every 119909 119910 isin 119881

2 sum119909119910isin119881

120583+119861(119909119910)

120583+119860(119909) and 120583+

119860(119910)

= 2 minus 2 sum119909119910isin119881

120583+119861(119909119910)

120583+119860(119909) and 120583+

119860(119910)

for every 119909 119910 isin 119881

(21)

Thus 119863minus(119866) = 2 minus 119863minus(119866) and 119863+(119866) = 2 minus 119863+(119866)Now

119863 (119866) + 119863 (119866) = (119863minus(119866) 119863

+(119866)) + (119863

minus(119866) 119863

+(119866))

= (119863minus(119866) + 119863

minus(119866)) (119863

+(119866) + 119863

+(119866))

= (2 2)

(22)

This completes the proof

Corollary 24 The complement of strictly balanced interval-valued fuzzy graph is strictly balanced

Theorem 25 Let 1198661and 119866

2be isomorphic interval-valued

fuzzy graphs If 1198662is balanced then 119866

1is balanced

4 Irregularity in Interval-ValuedFuzzy Graphs

Definition 26 Let119866 be an interval-valued fuzzy graph on119866lowastIf there is a vertex which is adjacent to vertices with distinctneighbourhood degrees then119866 is called an irregular interval-valued fuzzy graph That is deg(119909) = 119899 for all 119909 isin 119881

6 Journal of Applied Mathematics

1

23

[01 02] [01 02]

[02 03]

[02 06]

[03 04] [02 07]

Figure 2 Irregular interval-valued fuzzy graph

Example 27 Consider a graph 119866lowast such that

119881 = V1 V2 V3 119864 = V

1V2 V2V3 V1V3 (23)

Let 119860 be an interval-valued fuzzy subset of119881 and let 119861 be aninterval-valued fuzzy subset of 119864 sube 119881 times 119881 defined by

[[[

[

V1

V2

V3

120583minus119860

02 02 03

120583+119860

06 07 04

]]]

]

[[[

[

V1V2V1V3V2V3

120583minus119861

01 01 02

120583+119861

02 02 03

]]]

]

(24)

By routine computations we have deg(V1) = [05 11]

deg(V2) = [05 10] and deg(V

3) = [04 13] It is clear that

119866 as shown in Figure 2 is an irregular interval-valued fuzzygraph

Definition 28 Let 119866 be an interval-valued fuzzy graph Ifthere is a vertex which is adjacent to vertices with distinctclosed neighbourhood degrees then 119866 is called a totallyirregular interval-valued fuzzy graph

Example 29 Consider an interval-valued fuzzy graph119866 suchthat

119881 = V1 V2 V3 V4 V5

119864 = V1V2 V2V3 V2V4 V3V1 V3V4 V4V1 V4V5

(25)

By routine computations we have deg[V1] = [14 24]

deg[V2] = [14 24] deg[V

3] = [14 24] deg[V

4] = [16 26]

and deg[V5] = [06 08] It is clear from calculations that 119866 as

1 2

34

[04 06]

[04 06]

[02 02]

[03 05]

[03 07]

[01 02]

[02 02][01 04]

[0204]

[01 03]

[02 03]

[03 05]

5

Figure 3 119866 is totally irregular

1 2

4 3

[02 06]

[05 05]

[03 07]

[01 04] [01 04]

[04 04]

[01 02]

[01 02]

Figure 4 119866 is neighbourly irregular

shown in Figure 3 is a totally irregular interval-valued fuzzygraph

Definition 30 A connected interval-valued fuzzy graph 119866 issaid to be neighbourly irregular if every two adjacent verticesof 119866 have distinct open neighbourhood degree

Example 31 Consider an interval-valued fuzzy graph 119866 suchthat

119881 = V1 V2 V3 V4

119864 = V1V2 V2V3 V3V4 V4V1

(26)

By routine computations we have deg(V1) = [08 12]

deg(V2) = [06 10] deg(V

3) = [08 12] and deg(V

4) =

[06 10] Hence 119866 as shown in Figure 4 is neighbourlyirregular

Definition 32 A connected interval-valued fuzzy graph 119866 issaid to be neighbourly totally irregular if every two adjacentvertices of 119866 have distinct closed neighbourhood degree

Journal of Applied Mathematics 7

1 2

4 3

[03 06] [04 05]

[04 05] [02 07]

[01 02]

[01 02]

[01 04] [01 04]

Figure 5 119866 is neighbourly totally irregular

Example 33 Consider an interval-valued fuzzy graph119866 suchthat

119881 = V1 V2 V3 V4

119864 = V1V2 V2V3 V3V4 V4V1

(27)

By routine computations we have deg[V1] = [11 16]

deg[V2] = [09 18] deg[V

3] = [10 17] and deg[V

4] =

[09 18] Hence119866 as shown in Figure 5 is neighbourly totallyirregular

Definition 34 Let 119866 be a connected interval-valued fuzzygraph 119866 is called highly irregular if every vertex of 119866 isadjacent to vertices with distinct neighbourhood degrees

Example 35 Consider an interval-valued fuzzy graph119866 suchthat

119881 = V1 V2 V3 V4 V5 V6

119864 = V1V2 V2V3 V2V6 V3V4 V3V5 V4V5 V5V1

(28)

By routine computations we have deg(V1) = [04 08]

deg(V2) = [06 17] deg(V

3) = [09 13] deg(V

4) = [06 11]

deg(V5) = [10 18] and deg(V

6) = [01 04] Clearly 119866 as

shown in Figure 6 is highly irregular

Example 36 Consider an interval-valued fuzzy graph119866 suchthat

119881 = V1 V2 V3 V4

119864 = V1V2 V2V3 V3V4 V4V1

(29)

By routine computations we have deg(V1) = [06 10]

deg(V2) = [08 09] deg(V

3) = [06 10] and deg(V

4) =

1 2

4

35

[02 02]

[01 03]

[03 04]

[02

04

][02

04]

[05 05]

[02 06] [01 04]

[01 04]

[03 07]

[01 0

4]

[02 02]

[02 03]

6

Figure 6 119866 is highly irregular

[08 09]We see that every two adjacent vertices have distinctopen neighbourhood degree But the vertex V

2adjacent to

the vertices V1and V

3has the same neighbourhood degree

that is deg(V1) = deg(V

3) Hence 119866 as shown in Figure 7 is

neighbourly irregular but not highly irregular

Remark 37 A neighbourly irregular interval-valued fuzzygraph may not be highly irregular

Theorem38 An interval-valued fuzzy graph119866 is highly irreg-ular and neighbourly irregular interval-valued fuzzy graph ifand only if the neighbourhood degrees of all the vertices of 119866are distinct

Proof Let 119866 be an interval-valued fuzzy graph with 119899-vertices V

1 V2 V

119899 Assume that 119866 is highly irregular and

neighbourly irregular

Claim 1 The neighbourhood degrees of all vertices of 119866

are distinct Let deg(V119894) = [119896

119894 119897119894] 119894 = 1 2 119899 Let

the adjacent vertices of V1be V2 V3 V

119899with neighbour-

hood degrees [1198962 1198972] [1198963 1198973] [119896

119899 119897119899] respectively Then

1198962

= 1198963

= sdot sdot sdot = 119896119899

and 1198972

= 1198973

= sdot sdot sdot = 119897119899 since 119866 is highly

irregular Also 1198961

= 1198962

= 1198963

= sdot sdot sdot = 119896119899and 1198971

= 1198972

= 1198973

= sdot sdot sdot = 119897119899

since 119866 is neighbourly irregular Hence the neighbourhooddegree of all the vertices of 119866 is distinct

8 Journal of Applied Mathematics

1 2

34

[03 04]

[01 04]

[04 05]

[01 02]

[01 02]

[02 05]

[05 05]

[01 04]

Figure 7 119866 is neighbourly irregular but not highly irregular

Conversely assume that the neighbourhood degrees of allthe vertices of 119866 are distinct

Claim 2 119866 is highly irregular and neighbourly irregularinterval-valued fuzzy graph

Let deg(V119894) = [119896

119894 119897119894] 119894 = 1 2 119899 Given that

1198961

= 1198962

= 1198963

= sdot sdot sdot = 119896119899and 1198971

= 1198972

= 1198973

= sdot sdot sdot = 119897119899 which implies

that every two adjacent vertices have distinct neighbourhooddegrees and to every vertex the adjacent vertices have distinctneighbourhood degrees

Theorem 39 An interval-valued fuzzy graph 119866 of 119866lowast where119866lowast is a cycle with 3 vertices that is neighbourly irregular andhighly irregular if and only if the lower and upper membershipvalues of the vertices between every pair of vertices are alldistinct

Proof Assume that lower and upper membership values ofthe vertices are all distinct

Claim 1 119866 is neighbourly irregular and highly irregularinterval-valued fuzzy graph

Let V119894 V119895 V119896

isin 119881 Given that 120583minus119860(V119894) = 120583minus119860(V119895) =

120583minus

119860(V119896) and 120583

+

119860(V119894) = 120583+

119860(V119895) = 120583+

119860(V119896) which implies that

sum119909isin119873(119909)

120583minus119860(V119894) = sum119909isin119873(119909)

120583minus119860(V119895) = sum119909isin119873(119909)

120583minus119860(V119896) and

sum119909isin119873(119909)

120583+119860(V119894) = sum119909isin119873(119909)

120583+119860(V119895) = sum119909isin119873(119909)

120583+119860(V119896) That is

deg(V119894) = deg(V

119895) = deg(V

119896) Hence 119866 is neighbourly

irregular and highly irregularConversely assume that 119866 is neighbourly irregular and

highly irregular

Claim 2 Lower and upper membership values of the verticesare all distinct

Let deg(V119894) = [119896

119894 119897119894] 119894 = 1 2 119899 Suppose that lower

and upper membership value of any two vertices are the

1

23

[04 06]

[02 06]

[02 07] [02 06]

[04 06]

[04 06]

Figure 8 119866 is not neighbourly irregular but it is complete

same Let V1 V2

isin 119881 Let 120583minus119860(V1) = 120583minus

119860(V2) and 120583+

119860(V1) =

120583+119860(V2) Then deg(V

1) = deg(V

2) since 119866lowast is cycle which is

a contradiction to the fact that119866 is neighbourly irregular andhighly irregular interval-valued fuzzy graph Hence lowermembership and upper membership value of the vertices areall distinct

Remark 40 A complete interval-valued fuzzy graph may notbe neighbourly irregular

Example 41 Consider an interval-valued fuzzy graph 119866 suchthat

119881 = V1 V2 V3 119864 = V

1V2 V2V3 V1V3 (30)

By routine computations we have deg(V1) = [06 13]

deg(V2) = [06 13] and deg(V

3) = [08 12] We see that

neighbourhood degree of V1and V

2is not distinct Hence

119866 as shown in Figure 8 is not neighbourly irregular but it iscomplete

Remark 42 A neighbourly total irregular interval-valuedfuzzy graph may not be neighbourly irregular

Example 43 Consider an interval-valued fuzzy graph119866 suchthat

119881 = V1 V2 V3 V4 119864 = V

1V2 V2V3 V3V4 V4V1 (31)

By routine computations we have deg(V1) = [07 09]

deg(V2) = [08 11] deg(V

3) = [07 09] deg(V

4) = [08 11]

deg[V1] = [11 15] deg[V

2] = [11 15] deg[V

3] = [11 14]

and deg[V4] = [12 16] We see that deg[V

1] = deg[V

2]

Hence 119866 as shown in Figure 9 is neighbourly irregular butnot a neighbourly total irregular

Journal of Applied Mathematics 9

1 2

34

[04 06]

[01 04]

[04 05] [01 02]

[01 02]

[04 05]

[03 04]

[01 04]

Figure 9 119866 is neighbourly irregular but not neighbourly totalirregular

Proposition 44 If an interval-valued fuzzy graph 119866 is neigh-bourly irregular and [120583minus

119860 120583+119860] is a constant function then it is

neighbourly totally irregular

Proof Assume that 119866 is a neighbourly irregular interval-valued fuzzy graph Then the open neighbourhood degreesof every two adjacent vertices are distinct Let V

119894 V119895

isin 119881 beadjacent vertices with distinct open neighbourhood degrees[1198961 1198971] and [119896

2 1198972] where 119896

1= 1198962 1198971

= 1198972 Let us assume that

(1205831(V119894) ]1(V119894)) = (120583

1(V119895) ]1(V119895)) = [119888

1 1198882] where 119888

1 1198882are

constant and 1198881 1198882isin [0 1] Therefore deg

120583minus

[V119894] = deg

120583minus

(V119894) +

1205831(V119894) = 119896

1+ 1198881 deg120583+

[V119894] = deg

120583+

(V119894) + ]1(V119894) = 119897

1+

1198882deg120583minus

[V119895] = deg

120583minus

(V119895) + 1205831(V119895) = 1198962+ 1198881 and deg

120583+

[V119895] =

deg120583+

(V119895) + ]1(V119895) = 1198972+ 1198882

Claim Consider that deg120583minus

[V119894] = deg

120583minus

[V119895] and deg

120583+

[V119894] =

deg120583+

[V119895] Suppose that deg

120583minus

[V119894] = deg

120583minus

[V119895] and

deg120583+

[V119894] = deg

120583+

[V119895] Consider that

deg120583minus

[V119894] = deg

120583minus

[V119895]

1198961+ 1198881= 1198962+ 1198881

1198961minus 1198962= 1198881minus 1198881= 0

1198961= 1198962 which is a contradiction to 119896

1= 1198962

(32)

Therefore deg120583minus

[V119894] = deg

120583minus

[V119895] Similarly we consider that

deg120583+

[V119894] = deg

120583+

[V119895]

1198971+ 1198882= 1198972+ 1198882

1198971minus 1198972= 1198882minus 1198882= 0

1198971= 1198972 which is a contradiction to 119897

1= 1198972

(33)

Therefore deg120583+

[V119894] = deg

120583+

[V119895] Hence 119866 is a neighbourly

totally irregular interval-valued fuzzy graph

Theorem 45 If an interval-valued fuzzy graph 119866 is neigh-bourly totally irregular and [120583minus

119860 120583+119860] is a constant function

then it is a neighbourly irregular interval-valued fuzzy graph

Proof Assume that119866 is a neighbourly total irregular interval-valued fuzzy graph Then the closed neighbourhood degreeof every two adjacent vertices is distinct Let V

119894 V119895

isin 119881

and deg[V119894] = [119896

1 1198971] deg[V

119895] = [119896

2 1198972] where 119896

1= 1198962

and 1198971

= 1198972 Assume that (120583

1(V119894) ]1(V119894)) = [119888

1 1198882] and

(1205831(V119895) ]1(V119895)) = [119888

1 1198882] where 119888

1 1198882

isin [0 1] are constantand deg[V

119894] = deg[V

119895]

Claim Consider that deg(V119894) = deg(V

119895)

Given that deg[V119894] = deg[V

119895] which implies deg

120583minus

[V119894] =

deg120583minus

[V119895] and deg

120583+

[V119894] = deg

120583+

[V119895] now we consider that

deg120583minus

[V119894] = deg

120583minus

[V119895]

1198961+ 1198881

= 1198962+ 1198881

1198961

= 1198962

(34)

We now consider that

deg120583+

[V119894] = deg

120583+

[V119895]

1198971+ 1198882

= 1198972+ 1198882

1198971

= 1198972

(35)

That is the neighbourhood degrees of adjacent vertices of 119866are distinct Hence neighbourhood degree of every pair ofadjacent vertices is distinct in 119866

Proposition 46 If an interval-valued fuzzy graph 119866 is neigh-bourly irregular and neighbourly totally irregular then [120583

minus

119860 120583+

119860]

need not be a constant function

Remark 47 If 119866 is a neighbourly irregular interval-valuedfuzzy graph then interval-valued subgraph 119867 = (119860

1015840 1198611015840) of119866 may not be neighbourly irregular

Remark 48 If 119866 is a totally irregular interval-valued fuzzygraph then interval-valued fuzzy subgraph 119867 = (1198601015840 1198611015840) of119866 may not be totally irregular

5 Relationship between IFGs and IVFGs

In 2003 Deschrijver and Kerre [31] established the rela-tionships between some extensions of fuzzy sets In thissection we present the relationship between extensions offuzzy graphs Shannon and Atanassov [32] introduced thenotion of an intuitionistic fuzzy graph Some operations onintuitionistic fuzzy graphs are discussed in [33]

Definition 49 (see [32]) By an intuitionistic fuzzy graph (IFGfor short) 119866 of a graph 119866

lowast we mean a pair 119866 = (119860 119861)

10 Journal of Applied Mathematics

Fuzzy sets

Interval-valued

fuzzy sets

Intuitionisticfuzzy sets

Interval-valued

fuzzy graphs

Intuitionisticfuzzy graphs

Figure 10 Links between models

Interval-valued

fuzzy graphs

Intuitionistic

fuzzy graphs

Fuzzy graphs

Figure 11 Link between IFGs and IVFGs

where 119860 = (120583119860 ]119860) is an intuitionistic fuzzy set on 119881 and

119861 = (120583119861 ]119861) is an intuitionistic fuzzy relation on 119864 such that

120583119861(119909119910) le min (120583

119860(119909) 120583

119860(119910))

]119861(119909119910) le max (]

119860(119909) ]

119860(119910))

(36)

for all 119909119910 isin 119864 The class of all IFGs on 119866lowast will be denoted byIFG(119866lowast)

Theorem 50 (see [33]) If 1198661and 119866

2are intuitionistic fuzzy

graphs then 1198661cap 1198662 1198661cup 1198662 and 119866

1are intuitionistic fuzzy

graphs

Ju and Wang introduced the notion of interval-valuedfuzzy graph (IVFG for short) in [15] Some operations oninterval-valued fuzzy graphs are discussed in [18] The classof all IVFGs on 119866

lowast will be denoted byIVFG(119866lowast)

Theorem 51 (see [19]) If 1198661and 119866

2are interval-valued fuzzy

graphs then1198661cap11986621198661cup1198662 and119866

1are interval-valued fuzzy

graphs

Lemma 52 (IVFG(119866lowast) cup cap) and (IFG(119866lowast) cup cap) arecomplete lattices

Theorem 53 The mapping

1205951 IVFG (119866

lowast) 997888rarr IFG (119866

lowast) (37)

defined by

1205951(119861) = ((119909 120583

minus

119860(119909) 1 minus 120583

+

119860(119909)) | 119909 isin 119881

(119909119910 120583minus

119861(119909119910) 1 minus 120583

+

119861(119909119910)) | 119909119910 isin 119864)

(38)

where119861 = ((119909 120583

minus

119860(119909) 120583

+

119860(119909)) | 119909 isin 119881

(119909119910 120583minus

119861(119909119910) 120583

+

119861(119909119910)) | 119909119910 isin 119864)

(39)

is an isomorphism between lattices (IVFG(119866lowast) cup cap) and(IFG(119866lowast) cup cap)

Remark 54 From a pure mathematical point of viewTheorem 53 shows that the two concepts intuitionistic fuzzygraphs and interval-valued fuzzy graphs are equivalent

In Figures 10 and 11 we present the relationships that existbetween different models In these figures a double arrowbetween two theories means that they are equivalent a singlearrow 119883 rarr 119884 denotes that 119884 is an extension of 119883 InFigure 10 a dash arrow 119883[119884 denotes that model 119884 is basedon the previous model 119883

6 Conclusions

Graph theory has several interesting applications in systemanalysis operations research computer applications andeconomics Since most of the time the aspects of graphproblems are uncertain it is nice to deal with these aspectsvia the methods of fuzzy systems It is known that fuzzygraph theory has numerous applications in modern scienceand engineering especially in the field of information theoryneural networks expert systems cluster analysis medicaldiagnosis traffic engineering network routing town plan-ning and control theory Since interval-valued fuzzy settheory is an increasingly popular extension of fuzzy settheory where traditional [0 1]-valued membership degreesare replaced by intervals in [0 1] that approximate the(unknown) membership degrees specific types of interval-valued fuzzy graphs have been introduced and investigatedin this paper The natural extension of this research work isthe application of interval-valued fuzzy graphs in the area ofsoft computing including neural networks expert systemsdatabase theory and geographical information systems

Acknowledgments

The authors are thankful to the referees for their valuablecomments The authors are also thankful to Professor SyedMansoor Sarwar and Dr Faisal Aslam for their valuablesuggestions

References

[1] S G Shirinivas S Vetrivel and N M Elango ldquoApplications ofgraph theory in computer sciencemdashan overviewrdquo InternationalJournal of Engineering Science and Technology vol 2 no 9 pp4610ndash4621 2010

Journal of Applied Mathematics 11

[2] N Deo Graph Theory with Applications to Engineering andComputer Science Prentice Hall Englewood Cliffs NJ USA1990

[3] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoningmdashIrdquo Information Sciencesvol 8 pp 199ndash249 1975

[4] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[5] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 pp338ndash353 1965

[6] J M Mendel Uncertain Rule-Based Fuzzy Logic Systems Intro-duction and New Directions Prentice-Hall Upper Saddle RiverNJ USA 2001

[7] M B Gorzałczany ldquoA method of inference in approximatereasoning based on interval-valued fuzzy setsrdquo Fuzzy Sets andSystems vol 21 no 1 pp 1ndash17 1987

[8] M B Gorzałczany ldquoAn interval-valued fuzzy inferencemethodmdashsome basic propertiesrdquo Fuzzy Sets and Systems vol31 no 2 pp 243ndash251 1989

[9] M K Roy and R Biswas ldquoI-V fuzzy relations and Sanchezrsquosapproach for medical diagnosisrdquo Fuzzy Sets and Systems vol47 no 1 pp 35ndash38 1992

[10] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986

[11] A Kauffman Introduction a la Theorie des Sous-emsemblesFlous vol 1 Masson et Cie 1973

[12] A Rosenfeld ldquoFuzzy graphsrdquo in Fuzzy Sets and Their Applica-tions L A Zadeh K S Fu and M Shimura Eds pp 77ndash95Academic Press New York NY USA 1975

[13] J N Mordeson and C-S Peng ldquoOperations on fuzzy graphsrdquoInformation Sciences vol 79 no 3-4 pp 159ndash170 1994

[14] M S Sunitha and A V Kumar ldquoComplement of a fuzzy graphrdquoIndian Journal of Pure and Applied Mathematics vol 33 no 9pp 1451ndash1464 2002

[15] H Ju and L Wang ldquoInterval-valued fuzzy subsemigroupsand subgroups associated by intervalvalued fuzzy graphsrdquo inProceedings of the WRI Global Congress on Intelligent Systems(GCIS rsquo09) pp 484ndash487 Xiamen China May 2009

[16] MAkram ldquoBipolar fuzzy graphsrdquo Information Sciences vol 181no 24 pp 5548ndash5564 2011

[17] M Akram ldquoBipolar fuzzy graphs with applicationsrdquo KnowledgeBased Systems vol 39 pp 1ndash8 2013

[18] M Akram ldquoInterval-valued fuzzy line graphsrdquoNeural Comput-ing and Applications vol 21 pp 145ndash150 2012

[19] M Akram and W A Dudek ldquoInterval-valued fuzzy graphsrdquoComputers amp Mathematics with Applications vol 61 no 2 pp289ndash299 2011

[20] M Akram and B Davvaz ldquoStrong intuitionistic fuzzy graphsrdquoFilomat vol 26 no 1 pp 177ndash196 2012

[21] M Akram K H Dar and K P Shum ldquoInterval-valued (120572120573)-fuzzy K-algebrasrdquo Applied Soft Computing Journal vol 11 no 1pp 1213ndash1222 2011

[22] T AL-Hawary ldquoComplete fuzzy graphsrdquo International Journalof Mathematical Combinatorics vol 4 pp 26ndash34 2011

[23] P Bhattacharya ldquoSome remarks on fuzzy graphsrdquo PatternRecognition Letters vol 6 no 5 pp 297ndash302 1987

[24] S M Chen ldquoInterval-valued fuzzy hypergraph and fuzzypartitionrdquo IEEE Transactions on Systems Man and CyberneticsB vol 27 no 4 pp 725ndash733 1997

[25] G Deschrijver and C Cornelis ldquoRepresentability in interval-valued fuzzy set theoryrdquo International Journal of UncertaintyFuzziness and Knowledge-Based Systems vol 15 no 3 pp 345ndash361 2007

[26] A N Gani and S R Latha ldquoOn irregular fuzzy graphsrdquoAppliedMathematical Sciences vol 6 no 9ndash12 pp 517ndash523 2012

[27] X Ma J Zhan B Davvaz and Y B Jun ldquoSome kinds of (isin isinor119902)-interval-valued fuzzy ideals of BCI-algebrasrdquo InformationSciences vol 178 no 19 pp 3738ndash3754 2008

[28] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Studies in Fuzziness and SoftComputing PhysicaHeidelberg Germany 1998

[29] F Riaz and KM Ali ldquoApplications of graph theory in computersciencerdquo in Proceedings of the 3rd International Conferenceon Computational Intelligence Communication Systems andNetworks (CICSyN rsquo11) pp 142ndash145 Bali Indonesia July 2011

[30] M Behzad and G Chartrand ldquoNo graph is perfectrdquo TheAmerican Mathematical Monthly vol 74 pp 962ndash963 1967

[31] G Deschrijver and E E Kerre ldquoOn the relationship betweensome extensions of fuzzy set theoryrdquo Fuzzy Sets and Systemsvol 133 no 2 pp 227ndash235 2003

[32] A Shannon and K T Atanassov ldquoA first step to a theory ofthe intuitionistic fuzzy graphsrdquo in Proceeding of the FUBESTD Lakov Ed pp 59ndash61 Sofia Bulgaria 1994

[33] R Parvathi M G Karunambigai and K T Atanassov ldquoOper-ations on intuitionistic fuzzy graphsrdquo in Proceedings of theIEEE International Conference on Fuzzy Systems pp 1396ndash1401August 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Applied Mathematics

Definition 10 An interval-valued fuzzy graph 119866 = (119860 119861) iscalled complete if

120583minus

119861(119909119910) = min (120583

minus

119860(119909) 120583

minus

119860(119910))

120583+

119861(119909119910) = max (120583

+

119860(119909) 120583

+

119860(119910)) forall119909 119910 isin 119881

(6)

Example 11 Consider a graph119866lowast such that119881 = 119909 119910 119911 119864 =

119909119910 119910119911 119911119909 Let119860 be an interval-valued fuzzy subset of119881 andlet 119861 be an interval-valued fuzzy subset of 119864 defined by

120583minus

119860(119909) = 03 120583

minus

119860(119910) = 04 120583

minus

119860(119911) = 05

120583+

119860(119909) = 05 120583

+

119860(119910) = 07 120583

+

119860(119911) = 06

120583minus

119861(119909119910) = 03 120583

minus

119861(119910119911) = 04 120583

minus

119861(119911119909) = 03

120583+

119861(119909119910) = 05 120583

+

119861(119910119911) = 06 120583

+

119861(119911119909) = 05

(7)

Routine computations show that 119866 is both completeand totally regular interval-valued fuzzy graph but 119866 is notregular since deg(119909) = deg(119911) = deg(119910)

Theorem 12 Every complete interval-valued fuzzy graph istotally regular

Theorem 13 Let119866 = (119860 119861) be an interval-valued fuzzy graphof a graph 119866

lowast Then119860 = [120583minus119860 120583+119860] is a constant function if and

only if the following statements are equivalent

(a) 119866 is a regular interval-valued fuzzy graph(b) 119866 is a totally regular interval-valued fuzzy graph

Proof Suppose that 119860 = [120583minus119860 120583+119860] is a constant function Let

120583minus119860(119909) = 119888

1and 120583+

119860(119909) = 119888

2for all 119909 isin 119881

(a) rArr (b) Assume that 119866 is an 119899-regular interval-valuedfuzzy graph Then deg

120583minus

(119909) = 1198991and deg

120583+

(119909) = 1198992for all

119909 isin 119881 So

deg120583minus[119909] = deg

120583minus(119909) + 120583

minus

119860(119909)

deg120583+[119909] = deg

120583+(119909) + 120583

+

119860(119909) forall119909 isin 119881

(8)

Thus

deg120583minus[119909] = 119899

1+ 1198881 deg

120583+[119909] = 119899

2+ 1198882 forall119909 isin 119881 (9)

Hence 119866 is a totally regular interval-valued fuzzy graph(b) rArr (a) Suppose that 119866 is a totally regular interval-

valued fuzzy graph Then

deg120583minus[119909] = 119896

1 deg

120583+[119909] = 119896

2 forall119909 isin 119881 (10)

or

deg120583minus(119909) + 120583

minus

119860(119909) = 119896

1

deg120583+(119909) + 120583

+

119860(119909) = 119896

2 forall119909 isin 119881

(11)

or

deg120583minus(119909) + 119888

1= 1198961 deg

120583+(119909) + 119888

2= 1198962 forall119909 isin 119881 (12)

or

deg120583minus(119909) = 119896

1minus 1198881 deg

120583+(119909) = 119896

2minus 1198882 forall119909 isin 119881 (13)

Thus 119866 is a regular interval-valued fuzzy graph Hence (a)and (b) are equivalent

The converse part is obvious

Theorem 14 Let 119866 be an interval-valued fuzzy graph where acrisp graph 119866lowast is an odd cycle Then 119866 is a regular interval-valued fuzzy graph if and only if 119861 is a constant function

Proof If 119861 = [120583minus119861 120583+119861] is a constant function say 120583minus

119861= 1198881and

120583+119861

= 1198882for all 119909119910 isin 119864 then deg

120583minus

(119909) = 21198881and deg

120583+

(119909) =

21198882for every119909 isin 119881 Hence119866 is a regular interval-valued fuzzy

graphConversely suppose that 119866 is a (119896

1 1198962)-regular interval-

valued fuzzy graph Let 1198901 1198902 119890

2119899+1be the edges of 119866 in

that order Let 120583minus119861(1198901) = 1198881 120583minus119861(1198902) = 119896

1minus 1198881 120583minus119861(1198903) = 119896

1minus

(1198961minus 1198881) = 1198881 120583minus119861(1198904) = 1198961minus 1198881 and so on Therefore

120583minus

119861(119890119894) =

1198881 if 119894 is odd

1198961minus 1198881 if 119894 is even

(14)

Thus 120583minus119861(1198901) = 120583minus119861(1198902119899+1

) = 1198881 So if 119890

1and 1198902119899+1

incident at avertex V

1 then deg

120583minus

(V1) = 1198961 deg120583minus

(1198901) + deg

120583minus

(1198902119899+1

) = 1198961

1198881+ 1198881= 1198961 21198881= 1198961 and 119888

1= 11989612 This shows that 120583minus

119861is a

regular functionSimilarly let 120583+

119861(1198901) = 1198882 120583+119861(1198902) = 1198962minus 1198882 120583+119861(1198903) = 1198962minus

(1198962minus 1198882) = 1198882 120583+119861(1198904) = 1198962minus 1198882 and so on Therefore

120583+

119861(119890119894) =

1198882 if 119894 is odd

1198962minus 1198882 if 119894 is even

(15)

Thus 120583+119861(1198902) = 120583+

119861(1198902119899+1

) = 1198882 So if 119890

2and 1198902119899

incident ata vertex V

2 then deg

120583+

(V2) = 1198962 deg120583+

(1198902) + deg

120583+

(1198902119899+1

) =

1198962 1198882+ 1198882

= 1198962 21198882

= 1198962 and 119888

2= 11989622 This shows that

120583+119861is a constant function Hence 119861 = [120583minus

119861 120583+119861] is a constant

function

We state the following characterization without its proof

Theorem 15 Let 119866 be an interval-valued fuzzy graph where acrisp graph 119866lowast is an even cycle Then 119866 is a regular interval-valued fuzzy graph if and only if either 119861 = [120583minus

119861 120583+119861] is a

constant function or alternate edges have the samemembershipvalues

Definition 16 The density of an interval-valued fuzzygraphs 119866 is 119863(119866) = (119863

minus(119866) 119863+(119866)) where 119863minus(119866) =(2sum119909119910isin119881

(120583minus119861(119909119910)))(sum

119909119910isin119881(120583minus119860(119909) and 120583minus

119860(119910))) for 119909 119910 isin 119881

and119863+(119866) = (2sum119909119910isin119881

(120583+119861(119909119910)))(sum

119909119910isin119881(120583+119860(119909)and120583+

119860(119910))) for

119909 119910 isin 119881 An interval-valued fuzzy graph 119866 is balanced if119863(119867) le 119863(119866) that is 119863minus(119867) le 119863minus(119866) 119863+(119867) le 119863+(119866)

for all subgraphs119867 of 119866 An interval-valued fuzzy graph 119866 isstrictly balanced if for every 119909 119910 isin 119881 119863(119867) = 119863(119866) for allnonempty subgraphs

Journal of Applied Mathematics 5

Example 17 Consider the regular interval-valued fuzzy graph119866which is given in Example 7 Routine calculations show that119863minus(119866) = 067 and 119863+(119866) = 12 Thus 119863(119866) = (067 12)

Consider that 1198671

= 119886 119887 119888 1198672

= 119886 119888 1198673

= 119886 119889 and1198674

= 119887 119888 are a nonempty subgraphs of 119866 Then 119863(1198671) =

(067 12) 119863(1198672) = (0 0) 119863(119867

3) = (067 16) and 119863(119867

4) =

(067 16) It is easy to see that regular interval-valued fuzzygraph is not balanced

Remark 18 Every regular interval-valued fuzzy graph maynot be balanced

Example 19 Consider the regular interval-valued fuzzygraph 119866 which is given in Example 11 Routine calculationsshow that 119863minus(119866) = 2 and 119863+(119866) = 2 Thus 119863(119866) = (2 2)Consider119867

1= 119909 119910119867

2= 119909 119911119867

3= 119910 119911 be a nonempty

subgraphs of 119866 Then 119863(1198671) = (2 2) 119863(119867

2) = (2 2) and

119863(1198673) = (2 2) It is easy to see that complete interval-valued

fuzzy graph is balanced 119866 is also strictly balanced

Proposition 20 Any complete interval-valued fuzzy graph isbalanced

Proposition 21 Let 119866 be a self-complementary interval-valued fuzzy graph Then 119863(119866) = (1 1)

Proposition 22 Let 1198661and 119866

2be two balanced interval-

valued fuzzy graphs Then 1198661times 1198662is balanced if and only if

119863(1198661) = 119863(119866

2) = 119863(119866

1times 1198662)

Theorem23 Let119866 be a strictly balanced interval-valued fuzzygraph and let119866 be its complement then119863(119866)+119863(119866) = (2 2)

Proof Let 119866 be a strictly balanced interval-valued fuzzygraph and 119866 its complement Let119867 be a nonempty subgraphof 119866 Since 119866 is strictly balanced 119863(119866) = 119863(119867) for every119867 sube 119866 and 119909 119910 isin 119881 In 119866

120583minus119861(119909119910) = 120583

minus

119860(119909) and 120583

minus

119860(119910) minus 120583

minus

119861(119909119910) (16)

120583+119861(119909119910) = 120583

+

119860(119909) and 120583

+

119860(119910) minus 120583

+

119861(119909119910) (17)

for every 119909 119910 isin 119881 Dividing (16) by 120583minus119860(119909) and 120583minus

119860(119910) we get

120583minus119861(119909119910)

120583minus119860(119909) and 120583minus

119860(119910)

= 1 minus120583minus119861(119909119910)

120583minus119860(119909) and 120583minus

119860(119910)

for every 119909 119910isin119881

(18)

and dividing (17) by 120583+119860(119909) and 120583+

119860(119910) we get

120583+119861(119909119910)

120583+119860(119909) and 120583+

119860(119910)

= 1 minus120583+119861(119909119910)

120583+119860(119909) and 120583+

119860(119910)

for every 119909 119910isin119881

(19)

Then

sum119909119910isin119881

120583minus119861(119909119910)

120583minus119860(119909) and 120583minus

119860(119910)

= 1 minus sum119909119910isin119881

120583minus119861(119909119910)

120583minus119860(119909) and 120583minus

119860(119910)

for every 119909 119910 isin 119881

sum119909119910isin119881

120583+119861(119909119910)

120583+119860(119909) and 120583+

119860(119910)

= 1 minus sum119909119910isin119881

120583+119861(119909119910)

120583+119860(119909) and 120583+

119860(119910)

for every 119909 119910 isin 119881

(20)

Multiplying both sides the above equations by 2

2 sum119909119910isin119881

120583minus119861(119909119910)

120583minus119860(119909) and 120583minus

119860(119910)

= 2 minus 2 sum119909119910isin119881

120583minus119861(119909119910)

120583minus119860(119909) and 120583minus

119860(119910)

for every 119909 119910 isin 119881

2 sum119909119910isin119881

120583+119861(119909119910)

120583+119860(119909) and 120583+

119860(119910)

= 2 minus 2 sum119909119910isin119881

120583+119861(119909119910)

120583+119860(119909) and 120583+

119860(119910)

for every 119909 119910 isin 119881

(21)

Thus 119863minus(119866) = 2 minus 119863minus(119866) and 119863+(119866) = 2 minus 119863+(119866)Now

119863 (119866) + 119863 (119866) = (119863minus(119866) 119863

+(119866)) + (119863

minus(119866) 119863

+(119866))

= (119863minus(119866) + 119863

minus(119866)) (119863

+(119866) + 119863

+(119866))

= (2 2)

(22)

This completes the proof

Corollary 24 The complement of strictly balanced interval-valued fuzzy graph is strictly balanced

Theorem 25 Let 1198661and 119866

2be isomorphic interval-valued

fuzzy graphs If 1198662is balanced then 119866

1is balanced

4 Irregularity in Interval-ValuedFuzzy Graphs

Definition 26 Let119866 be an interval-valued fuzzy graph on119866lowastIf there is a vertex which is adjacent to vertices with distinctneighbourhood degrees then119866 is called an irregular interval-valued fuzzy graph That is deg(119909) = 119899 for all 119909 isin 119881

6 Journal of Applied Mathematics

1

23

[01 02] [01 02]

[02 03]

[02 06]

[03 04] [02 07]

Figure 2 Irregular interval-valued fuzzy graph

Example 27 Consider a graph 119866lowast such that

119881 = V1 V2 V3 119864 = V

1V2 V2V3 V1V3 (23)

Let 119860 be an interval-valued fuzzy subset of119881 and let 119861 be aninterval-valued fuzzy subset of 119864 sube 119881 times 119881 defined by

[[[

[

V1

V2

V3

120583minus119860

02 02 03

120583+119860

06 07 04

]]]

]

[[[

[

V1V2V1V3V2V3

120583minus119861

01 01 02

120583+119861

02 02 03

]]]

]

(24)

By routine computations we have deg(V1) = [05 11]

deg(V2) = [05 10] and deg(V

3) = [04 13] It is clear that

119866 as shown in Figure 2 is an irregular interval-valued fuzzygraph

Definition 28 Let 119866 be an interval-valued fuzzy graph Ifthere is a vertex which is adjacent to vertices with distinctclosed neighbourhood degrees then 119866 is called a totallyirregular interval-valued fuzzy graph

Example 29 Consider an interval-valued fuzzy graph119866 suchthat

119881 = V1 V2 V3 V4 V5

119864 = V1V2 V2V3 V2V4 V3V1 V3V4 V4V1 V4V5

(25)

By routine computations we have deg[V1] = [14 24]

deg[V2] = [14 24] deg[V

3] = [14 24] deg[V

4] = [16 26]

and deg[V5] = [06 08] It is clear from calculations that 119866 as

1 2

34

[04 06]

[04 06]

[02 02]

[03 05]

[03 07]

[01 02]

[02 02][01 04]

[0204]

[01 03]

[02 03]

[03 05]

5

Figure 3 119866 is totally irregular

1 2

4 3

[02 06]

[05 05]

[03 07]

[01 04] [01 04]

[04 04]

[01 02]

[01 02]

Figure 4 119866 is neighbourly irregular

shown in Figure 3 is a totally irregular interval-valued fuzzygraph

Definition 30 A connected interval-valued fuzzy graph 119866 issaid to be neighbourly irregular if every two adjacent verticesof 119866 have distinct open neighbourhood degree

Example 31 Consider an interval-valued fuzzy graph 119866 suchthat

119881 = V1 V2 V3 V4

119864 = V1V2 V2V3 V3V4 V4V1

(26)

By routine computations we have deg(V1) = [08 12]

deg(V2) = [06 10] deg(V

3) = [08 12] and deg(V

4) =

[06 10] Hence 119866 as shown in Figure 4 is neighbourlyirregular

Definition 32 A connected interval-valued fuzzy graph 119866 issaid to be neighbourly totally irregular if every two adjacentvertices of 119866 have distinct closed neighbourhood degree

Journal of Applied Mathematics 7

1 2

4 3

[03 06] [04 05]

[04 05] [02 07]

[01 02]

[01 02]

[01 04] [01 04]

Figure 5 119866 is neighbourly totally irregular

Example 33 Consider an interval-valued fuzzy graph119866 suchthat

119881 = V1 V2 V3 V4

119864 = V1V2 V2V3 V3V4 V4V1

(27)

By routine computations we have deg[V1] = [11 16]

deg[V2] = [09 18] deg[V

3] = [10 17] and deg[V

4] =

[09 18] Hence119866 as shown in Figure 5 is neighbourly totallyirregular

Definition 34 Let 119866 be a connected interval-valued fuzzygraph 119866 is called highly irregular if every vertex of 119866 isadjacent to vertices with distinct neighbourhood degrees

Example 35 Consider an interval-valued fuzzy graph119866 suchthat

119881 = V1 V2 V3 V4 V5 V6

119864 = V1V2 V2V3 V2V6 V3V4 V3V5 V4V5 V5V1

(28)

By routine computations we have deg(V1) = [04 08]

deg(V2) = [06 17] deg(V

3) = [09 13] deg(V

4) = [06 11]

deg(V5) = [10 18] and deg(V

6) = [01 04] Clearly 119866 as

shown in Figure 6 is highly irregular

Example 36 Consider an interval-valued fuzzy graph119866 suchthat

119881 = V1 V2 V3 V4

119864 = V1V2 V2V3 V3V4 V4V1

(29)

By routine computations we have deg(V1) = [06 10]

deg(V2) = [08 09] deg(V

3) = [06 10] and deg(V

4) =

1 2

4

35

[02 02]

[01 03]

[03 04]

[02

04

][02

04]

[05 05]

[02 06] [01 04]

[01 04]

[03 07]

[01 0

4]

[02 02]

[02 03]

6

Figure 6 119866 is highly irregular

[08 09]We see that every two adjacent vertices have distinctopen neighbourhood degree But the vertex V

2adjacent to

the vertices V1and V

3has the same neighbourhood degree

that is deg(V1) = deg(V

3) Hence 119866 as shown in Figure 7 is

neighbourly irregular but not highly irregular

Remark 37 A neighbourly irregular interval-valued fuzzygraph may not be highly irregular

Theorem38 An interval-valued fuzzy graph119866 is highly irreg-ular and neighbourly irregular interval-valued fuzzy graph ifand only if the neighbourhood degrees of all the vertices of 119866are distinct

Proof Let 119866 be an interval-valued fuzzy graph with 119899-vertices V

1 V2 V

119899 Assume that 119866 is highly irregular and

neighbourly irregular

Claim 1 The neighbourhood degrees of all vertices of 119866

are distinct Let deg(V119894) = [119896

119894 119897119894] 119894 = 1 2 119899 Let

the adjacent vertices of V1be V2 V3 V

119899with neighbour-

hood degrees [1198962 1198972] [1198963 1198973] [119896

119899 119897119899] respectively Then

1198962

= 1198963

= sdot sdot sdot = 119896119899

and 1198972

= 1198973

= sdot sdot sdot = 119897119899 since 119866 is highly

irregular Also 1198961

= 1198962

= 1198963

= sdot sdot sdot = 119896119899and 1198971

= 1198972

= 1198973

= sdot sdot sdot = 119897119899

since 119866 is neighbourly irregular Hence the neighbourhooddegree of all the vertices of 119866 is distinct

8 Journal of Applied Mathematics

1 2

34

[03 04]

[01 04]

[04 05]

[01 02]

[01 02]

[02 05]

[05 05]

[01 04]

Figure 7 119866 is neighbourly irregular but not highly irregular

Conversely assume that the neighbourhood degrees of allthe vertices of 119866 are distinct

Claim 2 119866 is highly irregular and neighbourly irregularinterval-valued fuzzy graph

Let deg(V119894) = [119896

119894 119897119894] 119894 = 1 2 119899 Given that

1198961

= 1198962

= 1198963

= sdot sdot sdot = 119896119899and 1198971

= 1198972

= 1198973

= sdot sdot sdot = 119897119899 which implies

that every two adjacent vertices have distinct neighbourhooddegrees and to every vertex the adjacent vertices have distinctneighbourhood degrees

Theorem 39 An interval-valued fuzzy graph 119866 of 119866lowast where119866lowast is a cycle with 3 vertices that is neighbourly irregular andhighly irregular if and only if the lower and upper membershipvalues of the vertices between every pair of vertices are alldistinct

Proof Assume that lower and upper membership values ofthe vertices are all distinct

Claim 1 119866 is neighbourly irregular and highly irregularinterval-valued fuzzy graph

Let V119894 V119895 V119896

isin 119881 Given that 120583minus119860(V119894) = 120583minus119860(V119895) =

120583minus

119860(V119896) and 120583

+

119860(V119894) = 120583+

119860(V119895) = 120583+

119860(V119896) which implies that

sum119909isin119873(119909)

120583minus119860(V119894) = sum119909isin119873(119909)

120583minus119860(V119895) = sum119909isin119873(119909)

120583minus119860(V119896) and

sum119909isin119873(119909)

120583+119860(V119894) = sum119909isin119873(119909)

120583+119860(V119895) = sum119909isin119873(119909)

120583+119860(V119896) That is

deg(V119894) = deg(V

119895) = deg(V

119896) Hence 119866 is neighbourly

irregular and highly irregularConversely assume that 119866 is neighbourly irregular and

highly irregular

Claim 2 Lower and upper membership values of the verticesare all distinct

Let deg(V119894) = [119896

119894 119897119894] 119894 = 1 2 119899 Suppose that lower

and upper membership value of any two vertices are the

1

23

[04 06]

[02 06]

[02 07] [02 06]

[04 06]

[04 06]

Figure 8 119866 is not neighbourly irregular but it is complete

same Let V1 V2

isin 119881 Let 120583minus119860(V1) = 120583minus

119860(V2) and 120583+

119860(V1) =

120583+119860(V2) Then deg(V

1) = deg(V

2) since 119866lowast is cycle which is

a contradiction to the fact that119866 is neighbourly irregular andhighly irregular interval-valued fuzzy graph Hence lowermembership and upper membership value of the vertices areall distinct

Remark 40 A complete interval-valued fuzzy graph may notbe neighbourly irregular

Example 41 Consider an interval-valued fuzzy graph 119866 suchthat

119881 = V1 V2 V3 119864 = V

1V2 V2V3 V1V3 (30)

By routine computations we have deg(V1) = [06 13]

deg(V2) = [06 13] and deg(V

3) = [08 12] We see that

neighbourhood degree of V1and V

2is not distinct Hence

119866 as shown in Figure 8 is not neighbourly irregular but it iscomplete

Remark 42 A neighbourly total irregular interval-valuedfuzzy graph may not be neighbourly irregular

Example 43 Consider an interval-valued fuzzy graph119866 suchthat

119881 = V1 V2 V3 V4 119864 = V

1V2 V2V3 V3V4 V4V1 (31)

By routine computations we have deg(V1) = [07 09]

deg(V2) = [08 11] deg(V

3) = [07 09] deg(V

4) = [08 11]

deg[V1] = [11 15] deg[V

2] = [11 15] deg[V

3] = [11 14]

and deg[V4] = [12 16] We see that deg[V

1] = deg[V

2]

Hence 119866 as shown in Figure 9 is neighbourly irregular butnot a neighbourly total irregular

Journal of Applied Mathematics 9

1 2

34

[04 06]

[01 04]

[04 05] [01 02]

[01 02]

[04 05]

[03 04]

[01 04]

Figure 9 119866 is neighbourly irregular but not neighbourly totalirregular

Proposition 44 If an interval-valued fuzzy graph 119866 is neigh-bourly irregular and [120583minus

119860 120583+119860] is a constant function then it is

neighbourly totally irregular

Proof Assume that 119866 is a neighbourly irregular interval-valued fuzzy graph Then the open neighbourhood degreesof every two adjacent vertices are distinct Let V

119894 V119895

isin 119881 beadjacent vertices with distinct open neighbourhood degrees[1198961 1198971] and [119896

2 1198972] where 119896

1= 1198962 1198971

= 1198972 Let us assume that

(1205831(V119894) ]1(V119894)) = (120583

1(V119895) ]1(V119895)) = [119888

1 1198882] where 119888

1 1198882are

constant and 1198881 1198882isin [0 1] Therefore deg

120583minus

[V119894] = deg

120583minus

(V119894) +

1205831(V119894) = 119896

1+ 1198881 deg120583+

[V119894] = deg

120583+

(V119894) + ]1(V119894) = 119897

1+

1198882deg120583minus

[V119895] = deg

120583minus

(V119895) + 1205831(V119895) = 1198962+ 1198881 and deg

120583+

[V119895] =

deg120583+

(V119895) + ]1(V119895) = 1198972+ 1198882

Claim Consider that deg120583minus

[V119894] = deg

120583minus

[V119895] and deg

120583+

[V119894] =

deg120583+

[V119895] Suppose that deg

120583minus

[V119894] = deg

120583minus

[V119895] and

deg120583+

[V119894] = deg

120583+

[V119895] Consider that

deg120583minus

[V119894] = deg

120583minus

[V119895]

1198961+ 1198881= 1198962+ 1198881

1198961minus 1198962= 1198881minus 1198881= 0

1198961= 1198962 which is a contradiction to 119896

1= 1198962

(32)

Therefore deg120583minus

[V119894] = deg

120583minus

[V119895] Similarly we consider that

deg120583+

[V119894] = deg

120583+

[V119895]

1198971+ 1198882= 1198972+ 1198882

1198971minus 1198972= 1198882minus 1198882= 0

1198971= 1198972 which is a contradiction to 119897

1= 1198972

(33)

Therefore deg120583+

[V119894] = deg

120583+

[V119895] Hence 119866 is a neighbourly

totally irregular interval-valued fuzzy graph

Theorem 45 If an interval-valued fuzzy graph 119866 is neigh-bourly totally irregular and [120583minus

119860 120583+119860] is a constant function

then it is a neighbourly irregular interval-valued fuzzy graph

Proof Assume that119866 is a neighbourly total irregular interval-valued fuzzy graph Then the closed neighbourhood degreeof every two adjacent vertices is distinct Let V

119894 V119895

isin 119881

and deg[V119894] = [119896

1 1198971] deg[V

119895] = [119896

2 1198972] where 119896

1= 1198962

and 1198971

= 1198972 Assume that (120583

1(V119894) ]1(V119894)) = [119888

1 1198882] and

(1205831(V119895) ]1(V119895)) = [119888

1 1198882] where 119888

1 1198882

isin [0 1] are constantand deg[V

119894] = deg[V

119895]

Claim Consider that deg(V119894) = deg(V

119895)

Given that deg[V119894] = deg[V

119895] which implies deg

120583minus

[V119894] =

deg120583minus

[V119895] and deg

120583+

[V119894] = deg

120583+

[V119895] now we consider that

deg120583minus

[V119894] = deg

120583minus

[V119895]

1198961+ 1198881

= 1198962+ 1198881

1198961

= 1198962

(34)

We now consider that

deg120583+

[V119894] = deg

120583+

[V119895]

1198971+ 1198882

= 1198972+ 1198882

1198971

= 1198972

(35)

That is the neighbourhood degrees of adjacent vertices of 119866are distinct Hence neighbourhood degree of every pair ofadjacent vertices is distinct in 119866

Proposition 46 If an interval-valued fuzzy graph 119866 is neigh-bourly irregular and neighbourly totally irregular then [120583

minus

119860 120583+

119860]

need not be a constant function

Remark 47 If 119866 is a neighbourly irregular interval-valuedfuzzy graph then interval-valued subgraph 119867 = (119860

1015840 1198611015840) of119866 may not be neighbourly irregular

Remark 48 If 119866 is a totally irregular interval-valued fuzzygraph then interval-valued fuzzy subgraph 119867 = (1198601015840 1198611015840) of119866 may not be totally irregular

5 Relationship between IFGs and IVFGs

In 2003 Deschrijver and Kerre [31] established the rela-tionships between some extensions of fuzzy sets In thissection we present the relationship between extensions offuzzy graphs Shannon and Atanassov [32] introduced thenotion of an intuitionistic fuzzy graph Some operations onintuitionistic fuzzy graphs are discussed in [33]

Definition 49 (see [32]) By an intuitionistic fuzzy graph (IFGfor short) 119866 of a graph 119866

lowast we mean a pair 119866 = (119860 119861)

10 Journal of Applied Mathematics

Fuzzy sets

Interval-valued

fuzzy sets

Intuitionisticfuzzy sets

Interval-valued

fuzzy graphs

Intuitionisticfuzzy graphs

Figure 10 Links between models

Interval-valued

fuzzy graphs

Intuitionistic

fuzzy graphs

Fuzzy graphs

Figure 11 Link between IFGs and IVFGs

where 119860 = (120583119860 ]119860) is an intuitionistic fuzzy set on 119881 and

119861 = (120583119861 ]119861) is an intuitionistic fuzzy relation on 119864 such that

120583119861(119909119910) le min (120583

119860(119909) 120583

119860(119910))

]119861(119909119910) le max (]

119860(119909) ]

119860(119910))

(36)

for all 119909119910 isin 119864 The class of all IFGs on 119866lowast will be denoted byIFG(119866lowast)

Theorem 50 (see [33]) If 1198661and 119866

2are intuitionistic fuzzy

graphs then 1198661cap 1198662 1198661cup 1198662 and 119866

1are intuitionistic fuzzy

graphs

Ju and Wang introduced the notion of interval-valuedfuzzy graph (IVFG for short) in [15] Some operations oninterval-valued fuzzy graphs are discussed in [18] The classof all IVFGs on 119866

lowast will be denoted byIVFG(119866lowast)

Theorem 51 (see [19]) If 1198661and 119866

2are interval-valued fuzzy

graphs then1198661cap11986621198661cup1198662 and119866

1are interval-valued fuzzy

graphs

Lemma 52 (IVFG(119866lowast) cup cap) and (IFG(119866lowast) cup cap) arecomplete lattices

Theorem 53 The mapping

1205951 IVFG (119866

lowast) 997888rarr IFG (119866

lowast) (37)

defined by

1205951(119861) = ((119909 120583

minus

119860(119909) 1 minus 120583

+

119860(119909)) | 119909 isin 119881

(119909119910 120583minus

119861(119909119910) 1 minus 120583

+

119861(119909119910)) | 119909119910 isin 119864)

(38)

where119861 = ((119909 120583

minus

119860(119909) 120583

+

119860(119909)) | 119909 isin 119881

(119909119910 120583minus

119861(119909119910) 120583

+

119861(119909119910)) | 119909119910 isin 119864)

(39)

is an isomorphism between lattices (IVFG(119866lowast) cup cap) and(IFG(119866lowast) cup cap)

Remark 54 From a pure mathematical point of viewTheorem 53 shows that the two concepts intuitionistic fuzzygraphs and interval-valued fuzzy graphs are equivalent

In Figures 10 and 11 we present the relationships that existbetween different models In these figures a double arrowbetween two theories means that they are equivalent a singlearrow 119883 rarr 119884 denotes that 119884 is an extension of 119883 InFigure 10 a dash arrow 119883[119884 denotes that model 119884 is basedon the previous model 119883

6 Conclusions

Graph theory has several interesting applications in systemanalysis operations research computer applications andeconomics Since most of the time the aspects of graphproblems are uncertain it is nice to deal with these aspectsvia the methods of fuzzy systems It is known that fuzzygraph theory has numerous applications in modern scienceand engineering especially in the field of information theoryneural networks expert systems cluster analysis medicaldiagnosis traffic engineering network routing town plan-ning and control theory Since interval-valued fuzzy settheory is an increasingly popular extension of fuzzy settheory where traditional [0 1]-valued membership degreesare replaced by intervals in [0 1] that approximate the(unknown) membership degrees specific types of interval-valued fuzzy graphs have been introduced and investigatedin this paper The natural extension of this research work isthe application of interval-valued fuzzy graphs in the area ofsoft computing including neural networks expert systemsdatabase theory and geographical information systems

Acknowledgments

The authors are thankful to the referees for their valuablecomments The authors are also thankful to Professor SyedMansoor Sarwar and Dr Faisal Aslam for their valuablesuggestions

References

[1] S G Shirinivas S Vetrivel and N M Elango ldquoApplications ofgraph theory in computer sciencemdashan overviewrdquo InternationalJournal of Engineering Science and Technology vol 2 no 9 pp4610ndash4621 2010

Journal of Applied Mathematics 11

[2] N Deo Graph Theory with Applications to Engineering andComputer Science Prentice Hall Englewood Cliffs NJ USA1990

[3] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoningmdashIrdquo Information Sciencesvol 8 pp 199ndash249 1975

[4] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[5] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 pp338ndash353 1965

[6] J M Mendel Uncertain Rule-Based Fuzzy Logic Systems Intro-duction and New Directions Prentice-Hall Upper Saddle RiverNJ USA 2001

[7] M B Gorzałczany ldquoA method of inference in approximatereasoning based on interval-valued fuzzy setsrdquo Fuzzy Sets andSystems vol 21 no 1 pp 1ndash17 1987

[8] M B Gorzałczany ldquoAn interval-valued fuzzy inferencemethodmdashsome basic propertiesrdquo Fuzzy Sets and Systems vol31 no 2 pp 243ndash251 1989

[9] M K Roy and R Biswas ldquoI-V fuzzy relations and Sanchezrsquosapproach for medical diagnosisrdquo Fuzzy Sets and Systems vol47 no 1 pp 35ndash38 1992

[10] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986

[11] A Kauffman Introduction a la Theorie des Sous-emsemblesFlous vol 1 Masson et Cie 1973

[12] A Rosenfeld ldquoFuzzy graphsrdquo in Fuzzy Sets and Their Applica-tions L A Zadeh K S Fu and M Shimura Eds pp 77ndash95Academic Press New York NY USA 1975

[13] J N Mordeson and C-S Peng ldquoOperations on fuzzy graphsrdquoInformation Sciences vol 79 no 3-4 pp 159ndash170 1994

[14] M S Sunitha and A V Kumar ldquoComplement of a fuzzy graphrdquoIndian Journal of Pure and Applied Mathematics vol 33 no 9pp 1451ndash1464 2002

[15] H Ju and L Wang ldquoInterval-valued fuzzy subsemigroupsand subgroups associated by intervalvalued fuzzy graphsrdquo inProceedings of the WRI Global Congress on Intelligent Systems(GCIS rsquo09) pp 484ndash487 Xiamen China May 2009

[16] MAkram ldquoBipolar fuzzy graphsrdquo Information Sciences vol 181no 24 pp 5548ndash5564 2011

[17] M Akram ldquoBipolar fuzzy graphs with applicationsrdquo KnowledgeBased Systems vol 39 pp 1ndash8 2013

[18] M Akram ldquoInterval-valued fuzzy line graphsrdquoNeural Comput-ing and Applications vol 21 pp 145ndash150 2012

[19] M Akram and W A Dudek ldquoInterval-valued fuzzy graphsrdquoComputers amp Mathematics with Applications vol 61 no 2 pp289ndash299 2011

[20] M Akram and B Davvaz ldquoStrong intuitionistic fuzzy graphsrdquoFilomat vol 26 no 1 pp 177ndash196 2012

[21] M Akram K H Dar and K P Shum ldquoInterval-valued (120572120573)-fuzzy K-algebrasrdquo Applied Soft Computing Journal vol 11 no 1pp 1213ndash1222 2011

[22] T AL-Hawary ldquoComplete fuzzy graphsrdquo International Journalof Mathematical Combinatorics vol 4 pp 26ndash34 2011

[23] P Bhattacharya ldquoSome remarks on fuzzy graphsrdquo PatternRecognition Letters vol 6 no 5 pp 297ndash302 1987

[24] S M Chen ldquoInterval-valued fuzzy hypergraph and fuzzypartitionrdquo IEEE Transactions on Systems Man and CyberneticsB vol 27 no 4 pp 725ndash733 1997

[25] G Deschrijver and C Cornelis ldquoRepresentability in interval-valued fuzzy set theoryrdquo International Journal of UncertaintyFuzziness and Knowledge-Based Systems vol 15 no 3 pp 345ndash361 2007

[26] A N Gani and S R Latha ldquoOn irregular fuzzy graphsrdquoAppliedMathematical Sciences vol 6 no 9ndash12 pp 517ndash523 2012

[27] X Ma J Zhan B Davvaz and Y B Jun ldquoSome kinds of (isin isinor119902)-interval-valued fuzzy ideals of BCI-algebrasrdquo InformationSciences vol 178 no 19 pp 3738ndash3754 2008

[28] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Studies in Fuzziness and SoftComputing PhysicaHeidelberg Germany 1998

[29] F Riaz and KM Ali ldquoApplications of graph theory in computersciencerdquo in Proceedings of the 3rd International Conferenceon Computational Intelligence Communication Systems andNetworks (CICSyN rsquo11) pp 142ndash145 Bali Indonesia July 2011

[30] M Behzad and G Chartrand ldquoNo graph is perfectrdquo TheAmerican Mathematical Monthly vol 74 pp 962ndash963 1967

[31] G Deschrijver and E E Kerre ldquoOn the relationship betweensome extensions of fuzzy set theoryrdquo Fuzzy Sets and Systemsvol 133 no 2 pp 227ndash235 2003

[32] A Shannon and K T Atanassov ldquoA first step to a theory ofthe intuitionistic fuzzy graphsrdquo in Proceeding of the FUBESTD Lakov Ed pp 59ndash61 Sofia Bulgaria 1994

[33] R Parvathi M G Karunambigai and K T Atanassov ldquoOper-ations on intuitionistic fuzzy graphsrdquo in Proceedings of theIEEE International Conference on Fuzzy Systems pp 1396ndash1401August 2009

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 5

Example 17 Consider the regular interval-valued fuzzy graph119866which is given in Example 7 Routine calculations show that119863minus(119866) = 067 and 119863+(119866) = 12 Thus 119863(119866) = (067 12)

Consider that 1198671

= 119886 119887 119888 1198672

= 119886 119888 1198673

= 119886 119889 and1198674

= 119887 119888 are a nonempty subgraphs of 119866 Then 119863(1198671) =

(067 12) 119863(1198672) = (0 0) 119863(119867

3) = (067 16) and 119863(119867

4) =

(067 16) It is easy to see that regular interval-valued fuzzygraph is not balanced

Remark 18 Every regular interval-valued fuzzy graph maynot be balanced

Example 19 Consider the regular interval-valued fuzzygraph 119866 which is given in Example 11 Routine calculationsshow that 119863minus(119866) = 2 and 119863+(119866) = 2 Thus 119863(119866) = (2 2)Consider119867

1= 119909 119910119867

2= 119909 119911119867

3= 119910 119911 be a nonempty

subgraphs of 119866 Then 119863(1198671) = (2 2) 119863(119867

2) = (2 2) and

119863(1198673) = (2 2) It is easy to see that complete interval-valued

fuzzy graph is balanced 119866 is also strictly balanced

Proposition 20 Any complete interval-valued fuzzy graph isbalanced

Proposition 21 Let 119866 be a self-complementary interval-valued fuzzy graph Then 119863(119866) = (1 1)

Proposition 22 Let 1198661and 119866

2be two balanced interval-

valued fuzzy graphs Then 1198661times 1198662is balanced if and only if

119863(1198661) = 119863(119866

2) = 119863(119866

1times 1198662)

Theorem23 Let119866 be a strictly balanced interval-valued fuzzygraph and let119866 be its complement then119863(119866)+119863(119866) = (2 2)

Proof Let 119866 be a strictly balanced interval-valued fuzzygraph and 119866 its complement Let119867 be a nonempty subgraphof 119866 Since 119866 is strictly balanced 119863(119866) = 119863(119867) for every119867 sube 119866 and 119909 119910 isin 119881 In 119866

120583minus119861(119909119910) = 120583

minus

119860(119909) and 120583

minus

119860(119910) minus 120583

minus

119861(119909119910) (16)

120583+119861(119909119910) = 120583

+

119860(119909) and 120583

+

119860(119910) minus 120583

+

119861(119909119910) (17)

for every 119909 119910 isin 119881 Dividing (16) by 120583minus119860(119909) and 120583minus

119860(119910) we get

120583minus119861(119909119910)

120583minus119860(119909) and 120583minus

119860(119910)

= 1 minus120583minus119861(119909119910)

120583minus119860(119909) and 120583minus

119860(119910)

for every 119909 119910isin119881

(18)

and dividing (17) by 120583+119860(119909) and 120583+

119860(119910) we get

120583+119861(119909119910)

120583+119860(119909) and 120583+

119860(119910)

= 1 minus120583+119861(119909119910)

120583+119860(119909) and 120583+

119860(119910)

for every 119909 119910isin119881

(19)

Then

sum119909119910isin119881

120583minus119861(119909119910)

120583minus119860(119909) and 120583minus

119860(119910)

= 1 minus sum119909119910isin119881

120583minus119861(119909119910)

120583minus119860(119909) and 120583minus

119860(119910)

for every 119909 119910 isin 119881

sum119909119910isin119881

120583+119861(119909119910)

120583+119860(119909) and 120583+

119860(119910)

= 1 minus sum119909119910isin119881

120583+119861(119909119910)

120583+119860(119909) and 120583+

119860(119910)

for every 119909 119910 isin 119881

(20)

Multiplying both sides the above equations by 2

2 sum119909119910isin119881

120583minus119861(119909119910)

120583minus119860(119909) and 120583minus

119860(119910)

= 2 minus 2 sum119909119910isin119881

120583minus119861(119909119910)

120583minus119860(119909) and 120583minus

119860(119910)

for every 119909 119910 isin 119881

2 sum119909119910isin119881

120583+119861(119909119910)

120583+119860(119909) and 120583+

119860(119910)

= 2 minus 2 sum119909119910isin119881

120583+119861(119909119910)

120583+119860(119909) and 120583+

119860(119910)

for every 119909 119910 isin 119881

(21)

Thus 119863minus(119866) = 2 minus 119863minus(119866) and 119863+(119866) = 2 minus 119863+(119866)Now

119863 (119866) + 119863 (119866) = (119863minus(119866) 119863

+(119866)) + (119863

minus(119866) 119863

+(119866))

= (119863minus(119866) + 119863

minus(119866)) (119863

+(119866) + 119863

+(119866))

= (2 2)

(22)

This completes the proof

Corollary 24 The complement of strictly balanced interval-valued fuzzy graph is strictly balanced

Theorem 25 Let 1198661and 119866

2be isomorphic interval-valued

fuzzy graphs If 1198662is balanced then 119866

1is balanced

4 Irregularity in Interval-ValuedFuzzy Graphs

Definition 26 Let119866 be an interval-valued fuzzy graph on119866lowastIf there is a vertex which is adjacent to vertices with distinctneighbourhood degrees then119866 is called an irregular interval-valued fuzzy graph That is deg(119909) = 119899 for all 119909 isin 119881

6 Journal of Applied Mathematics

1

23

[01 02] [01 02]

[02 03]

[02 06]

[03 04] [02 07]

Figure 2 Irregular interval-valued fuzzy graph

Example 27 Consider a graph 119866lowast such that

119881 = V1 V2 V3 119864 = V

1V2 V2V3 V1V3 (23)

Let 119860 be an interval-valued fuzzy subset of119881 and let 119861 be aninterval-valued fuzzy subset of 119864 sube 119881 times 119881 defined by

[[[

[

V1

V2

V3

120583minus119860

02 02 03

120583+119860

06 07 04

]]]

]

[[[

[

V1V2V1V3V2V3

120583minus119861

01 01 02

120583+119861

02 02 03

]]]

]

(24)

By routine computations we have deg(V1) = [05 11]

deg(V2) = [05 10] and deg(V

3) = [04 13] It is clear that

119866 as shown in Figure 2 is an irregular interval-valued fuzzygraph

Definition 28 Let 119866 be an interval-valued fuzzy graph Ifthere is a vertex which is adjacent to vertices with distinctclosed neighbourhood degrees then 119866 is called a totallyirregular interval-valued fuzzy graph

Example 29 Consider an interval-valued fuzzy graph119866 suchthat

119881 = V1 V2 V3 V4 V5

119864 = V1V2 V2V3 V2V4 V3V1 V3V4 V4V1 V4V5

(25)

By routine computations we have deg[V1] = [14 24]

deg[V2] = [14 24] deg[V

3] = [14 24] deg[V

4] = [16 26]

and deg[V5] = [06 08] It is clear from calculations that 119866 as

1 2

34

[04 06]

[04 06]

[02 02]

[03 05]

[03 07]

[01 02]

[02 02][01 04]

[0204]

[01 03]

[02 03]

[03 05]

5

Figure 3 119866 is totally irregular

1 2

4 3

[02 06]

[05 05]

[03 07]

[01 04] [01 04]

[04 04]

[01 02]

[01 02]

Figure 4 119866 is neighbourly irregular

shown in Figure 3 is a totally irregular interval-valued fuzzygraph

Definition 30 A connected interval-valued fuzzy graph 119866 issaid to be neighbourly irregular if every two adjacent verticesof 119866 have distinct open neighbourhood degree

Example 31 Consider an interval-valued fuzzy graph 119866 suchthat

119881 = V1 V2 V3 V4

119864 = V1V2 V2V3 V3V4 V4V1

(26)

By routine computations we have deg(V1) = [08 12]

deg(V2) = [06 10] deg(V

3) = [08 12] and deg(V

4) =

[06 10] Hence 119866 as shown in Figure 4 is neighbourlyirregular

Definition 32 A connected interval-valued fuzzy graph 119866 issaid to be neighbourly totally irregular if every two adjacentvertices of 119866 have distinct closed neighbourhood degree

Journal of Applied Mathematics 7

1 2

4 3

[03 06] [04 05]

[04 05] [02 07]

[01 02]

[01 02]

[01 04] [01 04]

Figure 5 119866 is neighbourly totally irregular

Example 33 Consider an interval-valued fuzzy graph119866 suchthat

119881 = V1 V2 V3 V4

119864 = V1V2 V2V3 V3V4 V4V1

(27)

By routine computations we have deg[V1] = [11 16]

deg[V2] = [09 18] deg[V

3] = [10 17] and deg[V

4] =

[09 18] Hence119866 as shown in Figure 5 is neighbourly totallyirregular

Definition 34 Let 119866 be a connected interval-valued fuzzygraph 119866 is called highly irregular if every vertex of 119866 isadjacent to vertices with distinct neighbourhood degrees

Example 35 Consider an interval-valued fuzzy graph119866 suchthat

119881 = V1 V2 V3 V4 V5 V6

119864 = V1V2 V2V3 V2V6 V3V4 V3V5 V4V5 V5V1

(28)

By routine computations we have deg(V1) = [04 08]

deg(V2) = [06 17] deg(V

3) = [09 13] deg(V

4) = [06 11]

deg(V5) = [10 18] and deg(V

6) = [01 04] Clearly 119866 as

shown in Figure 6 is highly irregular

Example 36 Consider an interval-valued fuzzy graph119866 suchthat

119881 = V1 V2 V3 V4

119864 = V1V2 V2V3 V3V4 V4V1

(29)

By routine computations we have deg(V1) = [06 10]

deg(V2) = [08 09] deg(V

3) = [06 10] and deg(V

4) =

1 2

4

35

[02 02]

[01 03]

[03 04]

[02

04

][02

04]

[05 05]

[02 06] [01 04]

[01 04]

[03 07]

[01 0

4]

[02 02]

[02 03]

6

Figure 6 119866 is highly irregular

[08 09]We see that every two adjacent vertices have distinctopen neighbourhood degree But the vertex V

2adjacent to

the vertices V1and V

3has the same neighbourhood degree

that is deg(V1) = deg(V

3) Hence 119866 as shown in Figure 7 is

neighbourly irregular but not highly irregular

Remark 37 A neighbourly irregular interval-valued fuzzygraph may not be highly irregular

Theorem38 An interval-valued fuzzy graph119866 is highly irreg-ular and neighbourly irregular interval-valued fuzzy graph ifand only if the neighbourhood degrees of all the vertices of 119866are distinct

Proof Let 119866 be an interval-valued fuzzy graph with 119899-vertices V

1 V2 V

119899 Assume that 119866 is highly irregular and

neighbourly irregular

Claim 1 The neighbourhood degrees of all vertices of 119866

are distinct Let deg(V119894) = [119896

119894 119897119894] 119894 = 1 2 119899 Let

the adjacent vertices of V1be V2 V3 V

119899with neighbour-

hood degrees [1198962 1198972] [1198963 1198973] [119896

119899 119897119899] respectively Then

1198962

= 1198963

= sdot sdot sdot = 119896119899

and 1198972

= 1198973

= sdot sdot sdot = 119897119899 since 119866 is highly

irregular Also 1198961

= 1198962

= 1198963

= sdot sdot sdot = 119896119899and 1198971

= 1198972

= 1198973

= sdot sdot sdot = 119897119899

since 119866 is neighbourly irregular Hence the neighbourhooddegree of all the vertices of 119866 is distinct

8 Journal of Applied Mathematics

1 2

34

[03 04]

[01 04]

[04 05]

[01 02]

[01 02]

[02 05]

[05 05]

[01 04]

Figure 7 119866 is neighbourly irregular but not highly irregular

Conversely assume that the neighbourhood degrees of allthe vertices of 119866 are distinct

Claim 2 119866 is highly irregular and neighbourly irregularinterval-valued fuzzy graph

Let deg(V119894) = [119896

119894 119897119894] 119894 = 1 2 119899 Given that

1198961

= 1198962

= 1198963

= sdot sdot sdot = 119896119899and 1198971

= 1198972

= 1198973

= sdot sdot sdot = 119897119899 which implies

that every two adjacent vertices have distinct neighbourhooddegrees and to every vertex the adjacent vertices have distinctneighbourhood degrees

Theorem 39 An interval-valued fuzzy graph 119866 of 119866lowast where119866lowast is a cycle with 3 vertices that is neighbourly irregular andhighly irregular if and only if the lower and upper membershipvalues of the vertices between every pair of vertices are alldistinct

Proof Assume that lower and upper membership values ofthe vertices are all distinct

Claim 1 119866 is neighbourly irregular and highly irregularinterval-valued fuzzy graph

Let V119894 V119895 V119896

isin 119881 Given that 120583minus119860(V119894) = 120583minus119860(V119895) =

120583minus

119860(V119896) and 120583

+

119860(V119894) = 120583+

119860(V119895) = 120583+

119860(V119896) which implies that

sum119909isin119873(119909)

120583minus119860(V119894) = sum119909isin119873(119909)

120583minus119860(V119895) = sum119909isin119873(119909)

120583minus119860(V119896) and

sum119909isin119873(119909)

120583+119860(V119894) = sum119909isin119873(119909)

120583+119860(V119895) = sum119909isin119873(119909)

120583+119860(V119896) That is

deg(V119894) = deg(V

119895) = deg(V

119896) Hence 119866 is neighbourly

irregular and highly irregularConversely assume that 119866 is neighbourly irregular and

highly irregular

Claim 2 Lower and upper membership values of the verticesare all distinct

Let deg(V119894) = [119896

119894 119897119894] 119894 = 1 2 119899 Suppose that lower

and upper membership value of any two vertices are the

1

23

[04 06]

[02 06]

[02 07] [02 06]

[04 06]

[04 06]

Figure 8 119866 is not neighbourly irregular but it is complete

same Let V1 V2

isin 119881 Let 120583minus119860(V1) = 120583minus

119860(V2) and 120583+

119860(V1) =

120583+119860(V2) Then deg(V

1) = deg(V

2) since 119866lowast is cycle which is

a contradiction to the fact that119866 is neighbourly irregular andhighly irregular interval-valued fuzzy graph Hence lowermembership and upper membership value of the vertices areall distinct

Remark 40 A complete interval-valued fuzzy graph may notbe neighbourly irregular

Example 41 Consider an interval-valued fuzzy graph 119866 suchthat

119881 = V1 V2 V3 119864 = V

1V2 V2V3 V1V3 (30)

By routine computations we have deg(V1) = [06 13]

deg(V2) = [06 13] and deg(V

3) = [08 12] We see that

neighbourhood degree of V1and V

2is not distinct Hence

119866 as shown in Figure 8 is not neighbourly irregular but it iscomplete

Remark 42 A neighbourly total irregular interval-valuedfuzzy graph may not be neighbourly irregular

Example 43 Consider an interval-valued fuzzy graph119866 suchthat

119881 = V1 V2 V3 V4 119864 = V

1V2 V2V3 V3V4 V4V1 (31)

By routine computations we have deg(V1) = [07 09]

deg(V2) = [08 11] deg(V

3) = [07 09] deg(V

4) = [08 11]

deg[V1] = [11 15] deg[V

2] = [11 15] deg[V

3] = [11 14]

and deg[V4] = [12 16] We see that deg[V

1] = deg[V

2]

Hence 119866 as shown in Figure 9 is neighbourly irregular butnot a neighbourly total irregular

Journal of Applied Mathematics 9

1 2

34

[04 06]

[01 04]

[04 05] [01 02]

[01 02]

[04 05]

[03 04]

[01 04]

Figure 9 119866 is neighbourly irregular but not neighbourly totalirregular

Proposition 44 If an interval-valued fuzzy graph 119866 is neigh-bourly irregular and [120583minus

119860 120583+119860] is a constant function then it is

neighbourly totally irregular

Proof Assume that 119866 is a neighbourly irregular interval-valued fuzzy graph Then the open neighbourhood degreesof every two adjacent vertices are distinct Let V

119894 V119895

isin 119881 beadjacent vertices with distinct open neighbourhood degrees[1198961 1198971] and [119896

2 1198972] where 119896

1= 1198962 1198971

= 1198972 Let us assume that

(1205831(V119894) ]1(V119894)) = (120583

1(V119895) ]1(V119895)) = [119888

1 1198882] where 119888

1 1198882are

constant and 1198881 1198882isin [0 1] Therefore deg

120583minus

[V119894] = deg

120583minus

(V119894) +

1205831(V119894) = 119896

1+ 1198881 deg120583+

[V119894] = deg

120583+

(V119894) + ]1(V119894) = 119897

1+

1198882deg120583minus

[V119895] = deg

120583minus

(V119895) + 1205831(V119895) = 1198962+ 1198881 and deg

120583+

[V119895] =

deg120583+

(V119895) + ]1(V119895) = 1198972+ 1198882

Claim Consider that deg120583minus

[V119894] = deg

120583minus

[V119895] and deg

120583+

[V119894] =

deg120583+

[V119895] Suppose that deg

120583minus

[V119894] = deg

120583minus

[V119895] and

deg120583+

[V119894] = deg

120583+

[V119895] Consider that

deg120583minus

[V119894] = deg

120583minus

[V119895]

1198961+ 1198881= 1198962+ 1198881

1198961minus 1198962= 1198881minus 1198881= 0

1198961= 1198962 which is a contradiction to 119896

1= 1198962

(32)

Therefore deg120583minus

[V119894] = deg

120583minus

[V119895] Similarly we consider that

deg120583+

[V119894] = deg

120583+

[V119895]

1198971+ 1198882= 1198972+ 1198882

1198971minus 1198972= 1198882minus 1198882= 0

1198971= 1198972 which is a contradiction to 119897

1= 1198972

(33)

Therefore deg120583+

[V119894] = deg

120583+

[V119895] Hence 119866 is a neighbourly

totally irregular interval-valued fuzzy graph

Theorem 45 If an interval-valued fuzzy graph 119866 is neigh-bourly totally irregular and [120583minus

119860 120583+119860] is a constant function

then it is a neighbourly irregular interval-valued fuzzy graph

Proof Assume that119866 is a neighbourly total irregular interval-valued fuzzy graph Then the closed neighbourhood degreeof every two adjacent vertices is distinct Let V

119894 V119895

isin 119881

and deg[V119894] = [119896

1 1198971] deg[V

119895] = [119896

2 1198972] where 119896

1= 1198962

and 1198971

= 1198972 Assume that (120583

1(V119894) ]1(V119894)) = [119888

1 1198882] and

(1205831(V119895) ]1(V119895)) = [119888

1 1198882] where 119888

1 1198882

isin [0 1] are constantand deg[V

119894] = deg[V

119895]

Claim Consider that deg(V119894) = deg(V

119895)

Given that deg[V119894] = deg[V

119895] which implies deg

120583minus

[V119894] =

deg120583minus

[V119895] and deg

120583+

[V119894] = deg

120583+

[V119895] now we consider that

deg120583minus

[V119894] = deg

120583minus

[V119895]

1198961+ 1198881

= 1198962+ 1198881

1198961

= 1198962

(34)

We now consider that

deg120583+

[V119894] = deg

120583+

[V119895]

1198971+ 1198882

= 1198972+ 1198882

1198971

= 1198972

(35)

That is the neighbourhood degrees of adjacent vertices of 119866are distinct Hence neighbourhood degree of every pair ofadjacent vertices is distinct in 119866

Proposition 46 If an interval-valued fuzzy graph 119866 is neigh-bourly irregular and neighbourly totally irregular then [120583

minus

119860 120583+

119860]

need not be a constant function

Remark 47 If 119866 is a neighbourly irregular interval-valuedfuzzy graph then interval-valued subgraph 119867 = (119860

1015840 1198611015840) of119866 may not be neighbourly irregular

Remark 48 If 119866 is a totally irregular interval-valued fuzzygraph then interval-valued fuzzy subgraph 119867 = (1198601015840 1198611015840) of119866 may not be totally irregular

5 Relationship between IFGs and IVFGs

In 2003 Deschrijver and Kerre [31] established the rela-tionships between some extensions of fuzzy sets In thissection we present the relationship between extensions offuzzy graphs Shannon and Atanassov [32] introduced thenotion of an intuitionistic fuzzy graph Some operations onintuitionistic fuzzy graphs are discussed in [33]

Definition 49 (see [32]) By an intuitionistic fuzzy graph (IFGfor short) 119866 of a graph 119866

lowast we mean a pair 119866 = (119860 119861)

10 Journal of Applied Mathematics

Fuzzy sets

Interval-valued

fuzzy sets

Intuitionisticfuzzy sets

Interval-valued

fuzzy graphs

Intuitionisticfuzzy graphs

Figure 10 Links between models

Interval-valued

fuzzy graphs

Intuitionistic

fuzzy graphs

Fuzzy graphs

Figure 11 Link between IFGs and IVFGs

where 119860 = (120583119860 ]119860) is an intuitionistic fuzzy set on 119881 and

119861 = (120583119861 ]119861) is an intuitionistic fuzzy relation on 119864 such that

120583119861(119909119910) le min (120583

119860(119909) 120583

119860(119910))

]119861(119909119910) le max (]

119860(119909) ]

119860(119910))

(36)

for all 119909119910 isin 119864 The class of all IFGs on 119866lowast will be denoted byIFG(119866lowast)

Theorem 50 (see [33]) If 1198661and 119866

2are intuitionistic fuzzy

graphs then 1198661cap 1198662 1198661cup 1198662 and 119866

1are intuitionistic fuzzy

graphs

Ju and Wang introduced the notion of interval-valuedfuzzy graph (IVFG for short) in [15] Some operations oninterval-valued fuzzy graphs are discussed in [18] The classof all IVFGs on 119866

lowast will be denoted byIVFG(119866lowast)

Theorem 51 (see [19]) If 1198661and 119866

2are interval-valued fuzzy

graphs then1198661cap11986621198661cup1198662 and119866

1are interval-valued fuzzy

graphs

Lemma 52 (IVFG(119866lowast) cup cap) and (IFG(119866lowast) cup cap) arecomplete lattices

Theorem 53 The mapping

1205951 IVFG (119866

lowast) 997888rarr IFG (119866

lowast) (37)

defined by

1205951(119861) = ((119909 120583

minus

119860(119909) 1 minus 120583

+

119860(119909)) | 119909 isin 119881

(119909119910 120583minus

119861(119909119910) 1 minus 120583

+

119861(119909119910)) | 119909119910 isin 119864)

(38)

where119861 = ((119909 120583

minus

119860(119909) 120583

+

119860(119909)) | 119909 isin 119881

(119909119910 120583minus

119861(119909119910) 120583

+

119861(119909119910)) | 119909119910 isin 119864)

(39)

is an isomorphism between lattices (IVFG(119866lowast) cup cap) and(IFG(119866lowast) cup cap)

Remark 54 From a pure mathematical point of viewTheorem 53 shows that the two concepts intuitionistic fuzzygraphs and interval-valued fuzzy graphs are equivalent

In Figures 10 and 11 we present the relationships that existbetween different models In these figures a double arrowbetween two theories means that they are equivalent a singlearrow 119883 rarr 119884 denotes that 119884 is an extension of 119883 InFigure 10 a dash arrow 119883[119884 denotes that model 119884 is basedon the previous model 119883

6 Conclusions

Graph theory has several interesting applications in systemanalysis operations research computer applications andeconomics Since most of the time the aspects of graphproblems are uncertain it is nice to deal with these aspectsvia the methods of fuzzy systems It is known that fuzzygraph theory has numerous applications in modern scienceand engineering especially in the field of information theoryneural networks expert systems cluster analysis medicaldiagnosis traffic engineering network routing town plan-ning and control theory Since interval-valued fuzzy settheory is an increasingly popular extension of fuzzy settheory where traditional [0 1]-valued membership degreesare replaced by intervals in [0 1] that approximate the(unknown) membership degrees specific types of interval-valued fuzzy graphs have been introduced and investigatedin this paper The natural extension of this research work isthe application of interval-valued fuzzy graphs in the area ofsoft computing including neural networks expert systemsdatabase theory and geographical information systems

Acknowledgments

The authors are thankful to the referees for their valuablecomments The authors are also thankful to Professor SyedMansoor Sarwar and Dr Faisal Aslam for their valuablesuggestions

References

[1] S G Shirinivas S Vetrivel and N M Elango ldquoApplications ofgraph theory in computer sciencemdashan overviewrdquo InternationalJournal of Engineering Science and Technology vol 2 no 9 pp4610ndash4621 2010

Journal of Applied Mathematics 11

[2] N Deo Graph Theory with Applications to Engineering andComputer Science Prentice Hall Englewood Cliffs NJ USA1990

[3] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoningmdashIrdquo Information Sciencesvol 8 pp 199ndash249 1975

[4] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[5] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 pp338ndash353 1965

[6] J M Mendel Uncertain Rule-Based Fuzzy Logic Systems Intro-duction and New Directions Prentice-Hall Upper Saddle RiverNJ USA 2001

[7] M B Gorzałczany ldquoA method of inference in approximatereasoning based on interval-valued fuzzy setsrdquo Fuzzy Sets andSystems vol 21 no 1 pp 1ndash17 1987

[8] M B Gorzałczany ldquoAn interval-valued fuzzy inferencemethodmdashsome basic propertiesrdquo Fuzzy Sets and Systems vol31 no 2 pp 243ndash251 1989

[9] M K Roy and R Biswas ldquoI-V fuzzy relations and Sanchezrsquosapproach for medical diagnosisrdquo Fuzzy Sets and Systems vol47 no 1 pp 35ndash38 1992

[10] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986

[11] A Kauffman Introduction a la Theorie des Sous-emsemblesFlous vol 1 Masson et Cie 1973

[12] A Rosenfeld ldquoFuzzy graphsrdquo in Fuzzy Sets and Their Applica-tions L A Zadeh K S Fu and M Shimura Eds pp 77ndash95Academic Press New York NY USA 1975

[13] J N Mordeson and C-S Peng ldquoOperations on fuzzy graphsrdquoInformation Sciences vol 79 no 3-4 pp 159ndash170 1994

[14] M S Sunitha and A V Kumar ldquoComplement of a fuzzy graphrdquoIndian Journal of Pure and Applied Mathematics vol 33 no 9pp 1451ndash1464 2002

[15] H Ju and L Wang ldquoInterval-valued fuzzy subsemigroupsand subgroups associated by intervalvalued fuzzy graphsrdquo inProceedings of the WRI Global Congress on Intelligent Systems(GCIS rsquo09) pp 484ndash487 Xiamen China May 2009

[16] MAkram ldquoBipolar fuzzy graphsrdquo Information Sciences vol 181no 24 pp 5548ndash5564 2011

[17] M Akram ldquoBipolar fuzzy graphs with applicationsrdquo KnowledgeBased Systems vol 39 pp 1ndash8 2013

[18] M Akram ldquoInterval-valued fuzzy line graphsrdquoNeural Comput-ing and Applications vol 21 pp 145ndash150 2012

[19] M Akram and W A Dudek ldquoInterval-valued fuzzy graphsrdquoComputers amp Mathematics with Applications vol 61 no 2 pp289ndash299 2011

[20] M Akram and B Davvaz ldquoStrong intuitionistic fuzzy graphsrdquoFilomat vol 26 no 1 pp 177ndash196 2012

[21] M Akram K H Dar and K P Shum ldquoInterval-valued (120572120573)-fuzzy K-algebrasrdquo Applied Soft Computing Journal vol 11 no 1pp 1213ndash1222 2011

[22] T AL-Hawary ldquoComplete fuzzy graphsrdquo International Journalof Mathematical Combinatorics vol 4 pp 26ndash34 2011

[23] P Bhattacharya ldquoSome remarks on fuzzy graphsrdquo PatternRecognition Letters vol 6 no 5 pp 297ndash302 1987

[24] S M Chen ldquoInterval-valued fuzzy hypergraph and fuzzypartitionrdquo IEEE Transactions on Systems Man and CyberneticsB vol 27 no 4 pp 725ndash733 1997

[25] G Deschrijver and C Cornelis ldquoRepresentability in interval-valued fuzzy set theoryrdquo International Journal of UncertaintyFuzziness and Knowledge-Based Systems vol 15 no 3 pp 345ndash361 2007

[26] A N Gani and S R Latha ldquoOn irregular fuzzy graphsrdquoAppliedMathematical Sciences vol 6 no 9ndash12 pp 517ndash523 2012

[27] X Ma J Zhan B Davvaz and Y B Jun ldquoSome kinds of (isin isinor119902)-interval-valued fuzzy ideals of BCI-algebrasrdquo InformationSciences vol 178 no 19 pp 3738ndash3754 2008

[28] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Studies in Fuzziness and SoftComputing PhysicaHeidelberg Germany 1998

[29] F Riaz and KM Ali ldquoApplications of graph theory in computersciencerdquo in Proceedings of the 3rd International Conferenceon Computational Intelligence Communication Systems andNetworks (CICSyN rsquo11) pp 142ndash145 Bali Indonesia July 2011

[30] M Behzad and G Chartrand ldquoNo graph is perfectrdquo TheAmerican Mathematical Monthly vol 74 pp 962ndash963 1967

[31] G Deschrijver and E E Kerre ldquoOn the relationship betweensome extensions of fuzzy set theoryrdquo Fuzzy Sets and Systemsvol 133 no 2 pp 227ndash235 2003

[32] A Shannon and K T Atanassov ldquoA first step to a theory ofthe intuitionistic fuzzy graphsrdquo in Proceeding of the FUBESTD Lakov Ed pp 59ndash61 Sofia Bulgaria 1994

[33] R Parvathi M G Karunambigai and K T Atanassov ldquoOper-ations on intuitionistic fuzzy graphsrdquo in Proceedings of theIEEE International Conference on Fuzzy Systems pp 1396ndash1401August 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Applied Mathematics

1

23

[01 02] [01 02]

[02 03]

[02 06]

[03 04] [02 07]

Figure 2 Irregular interval-valued fuzzy graph

Example 27 Consider a graph 119866lowast such that

119881 = V1 V2 V3 119864 = V

1V2 V2V3 V1V3 (23)

Let 119860 be an interval-valued fuzzy subset of119881 and let 119861 be aninterval-valued fuzzy subset of 119864 sube 119881 times 119881 defined by

[[[

[

V1

V2

V3

120583minus119860

02 02 03

120583+119860

06 07 04

]]]

]

[[[

[

V1V2V1V3V2V3

120583minus119861

01 01 02

120583+119861

02 02 03

]]]

]

(24)

By routine computations we have deg(V1) = [05 11]

deg(V2) = [05 10] and deg(V

3) = [04 13] It is clear that

119866 as shown in Figure 2 is an irregular interval-valued fuzzygraph

Definition 28 Let 119866 be an interval-valued fuzzy graph Ifthere is a vertex which is adjacent to vertices with distinctclosed neighbourhood degrees then 119866 is called a totallyirregular interval-valued fuzzy graph

Example 29 Consider an interval-valued fuzzy graph119866 suchthat

119881 = V1 V2 V3 V4 V5

119864 = V1V2 V2V3 V2V4 V3V1 V3V4 V4V1 V4V5

(25)

By routine computations we have deg[V1] = [14 24]

deg[V2] = [14 24] deg[V

3] = [14 24] deg[V

4] = [16 26]

and deg[V5] = [06 08] It is clear from calculations that 119866 as

1 2

34

[04 06]

[04 06]

[02 02]

[03 05]

[03 07]

[01 02]

[02 02][01 04]

[0204]

[01 03]

[02 03]

[03 05]

5

Figure 3 119866 is totally irregular

1 2

4 3

[02 06]

[05 05]

[03 07]

[01 04] [01 04]

[04 04]

[01 02]

[01 02]

Figure 4 119866 is neighbourly irregular

shown in Figure 3 is a totally irregular interval-valued fuzzygraph

Definition 30 A connected interval-valued fuzzy graph 119866 issaid to be neighbourly irregular if every two adjacent verticesof 119866 have distinct open neighbourhood degree

Example 31 Consider an interval-valued fuzzy graph 119866 suchthat

119881 = V1 V2 V3 V4

119864 = V1V2 V2V3 V3V4 V4V1

(26)

By routine computations we have deg(V1) = [08 12]

deg(V2) = [06 10] deg(V

3) = [08 12] and deg(V

4) =

[06 10] Hence 119866 as shown in Figure 4 is neighbourlyirregular

Definition 32 A connected interval-valued fuzzy graph 119866 issaid to be neighbourly totally irregular if every two adjacentvertices of 119866 have distinct closed neighbourhood degree

Journal of Applied Mathematics 7

1 2

4 3

[03 06] [04 05]

[04 05] [02 07]

[01 02]

[01 02]

[01 04] [01 04]

Figure 5 119866 is neighbourly totally irregular

Example 33 Consider an interval-valued fuzzy graph119866 suchthat

119881 = V1 V2 V3 V4

119864 = V1V2 V2V3 V3V4 V4V1

(27)

By routine computations we have deg[V1] = [11 16]

deg[V2] = [09 18] deg[V

3] = [10 17] and deg[V

4] =

[09 18] Hence119866 as shown in Figure 5 is neighbourly totallyirregular

Definition 34 Let 119866 be a connected interval-valued fuzzygraph 119866 is called highly irregular if every vertex of 119866 isadjacent to vertices with distinct neighbourhood degrees

Example 35 Consider an interval-valued fuzzy graph119866 suchthat

119881 = V1 V2 V3 V4 V5 V6

119864 = V1V2 V2V3 V2V6 V3V4 V3V5 V4V5 V5V1

(28)

By routine computations we have deg(V1) = [04 08]

deg(V2) = [06 17] deg(V

3) = [09 13] deg(V

4) = [06 11]

deg(V5) = [10 18] and deg(V

6) = [01 04] Clearly 119866 as

shown in Figure 6 is highly irregular

Example 36 Consider an interval-valued fuzzy graph119866 suchthat

119881 = V1 V2 V3 V4

119864 = V1V2 V2V3 V3V4 V4V1

(29)

By routine computations we have deg(V1) = [06 10]

deg(V2) = [08 09] deg(V

3) = [06 10] and deg(V

4) =

1 2

4

35

[02 02]

[01 03]

[03 04]

[02

04

][02

04]

[05 05]

[02 06] [01 04]

[01 04]

[03 07]

[01 0

4]

[02 02]

[02 03]

6

Figure 6 119866 is highly irregular

[08 09]We see that every two adjacent vertices have distinctopen neighbourhood degree But the vertex V

2adjacent to

the vertices V1and V

3has the same neighbourhood degree

that is deg(V1) = deg(V

3) Hence 119866 as shown in Figure 7 is

neighbourly irregular but not highly irregular

Remark 37 A neighbourly irregular interval-valued fuzzygraph may not be highly irregular

Theorem38 An interval-valued fuzzy graph119866 is highly irreg-ular and neighbourly irregular interval-valued fuzzy graph ifand only if the neighbourhood degrees of all the vertices of 119866are distinct

Proof Let 119866 be an interval-valued fuzzy graph with 119899-vertices V

1 V2 V

119899 Assume that 119866 is highly irregular and

neighbourly irregular

Claim 1 The neighbourhood degrees of all vertices of 119866

are distinct Let deg(V119894) = [119896

119894 119897119894] 119894 = 1 2 119899 Let

the adjacent vertices of V1be V2 V3 V

119899with neighbour-

hood degrees [1198962 1198972] [1198963 1198973] [119896

119899 119897119899] respectively Then

1198962

= 1198963

= sdot sdot sdot = 119896119899

and 1198972

= 1198973

= sdot sdot sdot = 119897119899 since 119866 is highly

irregular Also 1198961

= 1198962

= 1198963

= sdot sdot sdot = 119896119899and 1198971

= 1198972

= 1198973

= sdot sdot sdot = 119897119899

since 119866 is neighbourly irregular Hence the neighbourhooddegree of all the vertices of 119866 is distinct

8 Journal of Applied Mathematics

1 2

34

[03 04]

[01 04]

[04 05]

[01 02]

[01 02]

[02 05]

[05 05]

[01 04]

Figure 7 119866 is neighbourly irregular but not highly irregular

Conversely assume that the neighbourhood degrees of allthe vertices of 119866 are distinct

Claim 2 119866 is highly irregular and neighbourly irregularinterval-valued fuzzy graph

Let deg(V119894) = [119896

119894 119897119894] 119894 = 1 2 119899 Given that

1198961

= 1198962

= 1198963

= sdot sdot sdot = 119896119899and 1198971

= 1198972

= 1198973

= sdot sdot sdot = 119897119899 which implies

that every two adjacent vertices have distinct neighbourhooddegrees and to every vertex the adjacent vertices have distinctneighbourhood degrees

Theorem 39 An interval-valued fuzzy graph 119866 of 119866lowast where119866lowast is a cycle with 3 vertices that is neighbourly irregular andhighly irregular if and only if the lower and upper membershipvalues of the vertices between every pair of vertices are alldistinct

Proof Assume that lower and upper membership values ofthe vertices are all distinct

Claim 1 119866 is neighbourly irregular and highly irregularinterval-valued fuzzy graph

Let V119894 V119895 V119896

isin 119881 Given that 120583minus119860(V119894) = 120583minus119860(V119895) =

120583minus

119860(V119896) and 120583

+

119860(V119894) = 120583+

119860(V119895) = 120583+

119860(V119896) which implies that

sum119909isin119873(119909)

120583minus119860(V119894) = sum119909isin119873(119909)

120583minus119860(V119895) = sum119909isin119873(119909)

120583minus119860(V119896) and

sum119909isin119873(119909)

120583+119860(V119894) = sum119909isin119873(119909)

120583+119860(V119895) = sum119909isin119873(119909)

120583+119860(V119896) That is

deg(V119894) = deg(V

119895) = deg(V

119896) Hence 119866 is neighbourly

irregular and highly irregularConversely assume that 119866 is neighbourly irregular and

highly irregular

Claim 2 Lower and upper membership values of the verticesare all distinct

Let deg(V119894) = [119896

119894 119897119894] 119894 = 1 2 119899 Suppose that lower

and upper membership value of any two vertices are the

1

23

[04 06]

[02 06]

[02 07] [02 06]

[04 06]

[04 06]

Figure 8 119866 is not neighbourly irregular but it is complete

same Let V1 V2

isin 119881 Let 120583minus119860(V1) = 120583minus

119860(V2) and 120583+

119860(V1) =

120583+119860(V2) Then deg(V

1) = deg(V

2) since 119866lowast is cycle which is

a contradiction to the fact that119866 is neighbourly irregular andhighly irregular interval-valued fuzzy graph Hence lowermembership and upper membership value of the vertices areall distinct

Remark 40 A complete interval-valued fuzzy graph may notbe neighbourly irregular

Example 41 Consider an interval-valued fuzzy graph 119866 suchthat

119881 = V1 V2 V3 119864 = V

1V2 V2V3 V1V3 (30)

By routine computations we have deg(V1) = [06 13]

deg(V2) = [06 13] and deg(V

3) = [08 12] We see that

neighbourhood degree of V1and V

2is not distinct Hence

119866 as shown in Figure 8 is not neighbourly irregular but it iscomplete

Remark 42 A neighbourly total irregular interval-valuedfuzzy graph may not be neighbourly irregular

Example 43 Consider an interval-valued fuzzy graph119866 suchthat

119881 = V1 V2 V3 V4 119864 = V

1V2 V2V3 V3V4 V4V1 (31)

By routine computations we have deg(V1) = [07 09]

deg(V2) = [08 11] deg(V

3) = [07 09] deg(V

4) = [08 11]

deg[V1] = [11 15] deg[V

2] = [11 15] deg[V

3] = [11 14]

and deg[V4] = [12 16] We see that deg[V

1] = deg[V

2]

Hence 119866 as shown in Figure 9 is neighbourly irregular butnot a neighbourly total irregular

Journal of Applied Mathematics 9

1 2

34

[04 06]

[01 04]

[04 05] [01 02]

[01 02]

[04 05]

[03 04]

[01 04]

Figure 9 119866 is neighbourly irregular but not neighbourly totalirregular

Proposition 44 If an interval-valued fuzzy graph 119866 is neigh-bourly irregular and [120583minus

119860 120583+119860] is a constant function then it is

neighbourly totally irregular

Proof Assume that 119866 is a neighbourly irregular interval-valued fuzzy graph Then the open neighbourhood degreesof every two adjacent vertices are distinct Let V

119894 V119895

isin 119881 beadjacent vertices with distinct open neighbourhood degrees[1198961 1198971] and [119896

2 1198972] where 119896

1= 1198962 1198971

= 1198972 Let us assume that

(1205831(V119894) ]1(V119894)) = (120583

1(V119895) ]1(V119895)) = [119888

1 1198882] where 119888

1 1198882are

constant and 1198881 1198882isin [0 1] Therefore deg

120583minus

[V119894] = deg

120583minus

(V119894) +

1205831(V119894) = 119896

1+ 1198881 deg120583+

[V119894] = deg

120583+

(V119894) + ]1(V119894) = 119897

1+

1198882deg120583minus

[V119895] = deg

120583minus

(V119895) + 1205831(V119895) = 1198962+ 1198881 and deg

120583+

[V119895] =

deg120583+

(V119895) + ]1(V119895) = 1198972+ 1198882

Claim Consider that deg120583minus

[V119894] = deg

120583minus

[V119895] and deg

120583+

[V119894] =

deg120583+

[V119895] Suppose that deg

120583minus

[V119894] = deg

120583minus

[V119895] and

deg120583+

[V119894] = deg

120583+

[V119895] Consider that

deg120583minus

[V119894] = deg

120583minus

[V119895]

1198961+ 1198881= 1198962+ 1198881

1198961minus 1198962= 1198881minus 1198881= 0

1198961= 1198962 which is a contradiction to 119896

1= 1198962

(32)

Therefore deg120583minus

[V119894] = deg

120583minus

[V119895] Similarly we consider that

deg120583+

[V119894] = deg

120583+

[V119895]

1198971+ 1198882= 1198972+ 1198882

1198971minus 1198972= 1198882minus 1198882= 0

1198971= 1198972 which is a contradiction to 119897

1= 1198972

(33)

Therefore deg120583+

[V119894] = deg

120583+

[V119895] Hence 119866 is a neighbourly

totally irregular interval-valued fuzzy graph

Theorem 45 If an interval-valued fuzzy graph 119866 is neigh-bourly totally irregular and [120583minus

119860 120583+119860] is a constant function

then it is a neighbourly irregular interval-valued fuzzy graph

Proof Assume that119866 is a neighbourly total irregular interval-valued fuzzy graph Then the closed neighbourhood degreeof every two adjacent vertices is distinct Let V

119894 V119895

isin 119881

and deg[V119894] = [119896

1 1198971] deg[V

119895] = [119896

2 1198972] where 119896

1= 1198962

and 1198971

= 1198972 Assume that (120583

1(V119894) ]1(V119894)) = [119888

1 1198882] and

(1205831(V119895) ]1(V119895)) = [119888

1 1198882] where 119888

1 1198882

isin [0 1] are constantand deg[V

119894] = deg[V

119895]

Claim Consider that deg(V119894) = deg(V

119895)

Given that deg[V119894] = deg[V

119895] which implies deg

120583minus

[V119894] =

deg120583minus

[V119895] and deg

120583+

[V119894] = deg

120583+

[V119895] now we consider that

deg120583minus

[V119894] = deg

120583minus

[V119895]

1198961+ 1198881

= 1198962+ 1198881

1198961

= 1198962

(34)

We now consider that

deg120583+

[V119894] = deg

120583+

[V119895]

1198971+ 1198882

= 1198972+ 1198882

1198971

= 1198972

(35)

That is the neighbourhood degrees of adjacent vertices of 119866are distinct Hence neighbourhood degree of every pair ofadjacent vertices is distinct in 119866

Proposition 46 If an interval-valued fuzzy graph 119866 is neigh-bourly irregular and neighbourly totally irregular then [120583

minus

119860 120583+

119860]

need not be a constant function

Remark 47 If 119866 is a neighbourly irregular interval-valuedfuzzy graph then interval-valued subgraph 119867 = (119860

1015840 1198611015840) of119866 may not be neighbourly irregular

Remark 48 If 119866 is a totally irregular interval-valued fuzzygraph then interval-valued fuzzy subgraph 119867 = (1198601015840 1198611015840) of119866 may not be totally irregular

5 Relationship between IFGs and IVFGs

In 2003 Deschrijver and Kerre [31] established the rela-tionships between some extensions of fuzzy sets In thissection we present the relationship between extensions offuzzy graphs Shannon and Atanassov [32] introduced thenotion of an intuitionistic fuzzy graph Some operations onintuitionistic fuzzy graphs are discussed in [33]

Definition 49 (see [32]) By an intuitionistic fuzzy graph (IFGfor short) 119866 of a graph 119866

lowast we mean a pair 119866 = (119860 119861)

10 Journal of Applied Mathematics

Fuzzy sets

Interval-valued

fuzzy sets

Intuitionisticfuzzy sets

Interval-valued

fuzzy graphs

Intuitionisticfuzzy graphs

Figure 10 Links between models

Interval-valued

fuzzy graphs

Intuitionistic

fuzzy graphs

Fuzzy graphs

Figure 11 Link between IFGs and IVFGs

where 119860 = (120583119860 ]119860) is an intuitionistic fuzzy set on 119881 and

119861 = (120583119861 ]119861) is an intuitionistic fuzzy relation on 119864 such that

120583119861(119909119910) le min (120583

119860(119909) 120583

119860(119910))

]119861(119909119910) le max (]

119860(119909) ]

119860(119910))

(36)

for all 119909119910 isin 119864 The class of all IFGs on 119866lowast will be denoted byIFG(119866lowast)

Theorem 50 (see [33]) If 1198661and 119866

2are intuitionistic fuzzy

graphs then 1198661cap 1198662 1198661cup 1198662 and 119866

1are intuitionistic fuzzy

graphs

Ju and Wang introduced the notion of interval-valuedfuzzy graph (IVFG for short) in [15] Some operations oninterval-valued fuzzy graphs are discussed in [18] The classof all IVFGs on 119866

lowast will be denoted byIVFG(119866lowast)

Theorem 51 (see [19]) If 1198661and 119866

2are interval-valued fuzzy

graphs then1198661cap11986621198661cup1198662 and119866

1are interval-valued fuzzy

graphs

Lemma 52 (IVFG(119866lowast) cup cap) and (IFG(119866lowast) cup cap) arecomplete lattices

Theorem 53 The mapping

1205951 IVFG (119866

lowast) 997888rarr IFG (119866

lowast) (37)

defined by

1205951(119861) = ((119909 120583

minus

119860(119909) 1 minus 120583

+

119860(119909)) | 119909 isin 119881

(119909119910 120583minus

119861(119909119910) 1 minus 120583

+

119861(119909119910)) | 119909119910 isin 119864)

(38)

where119861 = ((119909 120583

minus

119860(119909) 120583

+

119860(119909)) | 119909 isin 119881

(119909119910 120583minus

119861(119909119910) 120583

+

119861(119909119910)) | 119909119910 isin 119864)

(39)

is an isomorphism between lattices (IVFG(119866lowast) cup cap) and(IFG(119866lowast) cup cap)

Remark 54 From a pure mathematical point of viewTheorem 53 shows that the two concepts intuitionistic fuzzygraphs and interval-valued fuzzy graphs are equivalent

In Figures 10 and 11 we present the relationships that existbetween different models In these figures a double arrowbetween two theories means that they are equivalent a singlearrow 119883 rarr 119884 denotes that 119884 is an extension of 119883 InFigure 10 a dash arrow 119883[119884 denotes that model 119884 is basedon the previous model 119883

6 Conclusions

Graph theory has several interesting applications in systemanalysis operations research computer applications andeconomics Since most of the time the aspects of graphproblems are uncertain it is nice to deal with these aspectsvia the methods of fuzzy systems It is known that fuzzygraph theory has numerous applications in modern scienceand engineering especially in the field of information theoryneural networks expert systems cluster analysis medicaldiagnosis traffic engineering network routing town plan-ning and control theory Since interval-valued fuzzy settheory is an increasingly popular extension of fuzzy settheory where traditional [0 1]-valued membership degreesare replaced by intervals in [0 1] that approximate the(unknown) membership degrees specific types of interval-valued fuzzy graphs have been introduced and investigatedin this paper The natural extension of this research work isthe application of interval-valued fuzzy graphs in the area ofsoft computing including neural networks expert systemsdatabase theory and geographical information systems

Acknowledgments

The authors are thankful to the referees for their valuablecomments The authors are also thankful to Professor SyedMansoor Sarwar and Dr Faisal Aslam for their valuablesuggestions

References

[1] S G Shirinivas S Vetrivel and N M Elango ldquoApplications ofgraph theory in computer sciencemdashan overviewrdquo InternationalJournal of Engineering Science and Technology vol 2 no 9 pp4610ndash4621 2010

Journal of Applied Mathematics 11

[2] N Deo Graph Theory with Applications to Engineering andComputer Science Prentice Hall Englewood Cliffs NJ USA1990

[3] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoningmdashIrdquo Information Sciencesvol 8 pp 199ndash249 1975

[4] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[5] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 pp338ndash353 1965

[6] J M Mendel Uncertain Rule-Based Fuzzy Logic Systems Intro-duction and New Directions Prentice-Hall Upper Saddle RiverNJ USA 2001

[7] M B Gorzałczany ldquoA method of inference in approximatereasoning based on interval-valued fuzzy setsrdquo Fuzzy Sets andSystems vol 21 no 1 pp 1ndash17 1987

[8] M B Gorzałczany ldquoAn interval-valued fuzzy inferencemethodmdashsome basic propertiesrdquo Fuzzy Sets and Systems vol31 no 2 pp 243ndash251 1989

[9] M K Roy and R Biswas ldquoI-V fuzzy relations and Sanchezrsquosapproach for medical diagnosisrdquo Fuzzy Sets and Systems vol47 no 1 pp 35ndash38 1992

[10] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986

[11] A Kauffman Introduction a la Theorie des Sous-emsemblesFlous vol 1 Masson et Cie 1973

[12] A Rosenfeld ldquoFuzzy graphsrdquo in Fuzzy Sets and Their Applica-tions L A Zadeh K S Fu and M Shimura Eds pp 77ndash95Academic Press New York NY USA 1975

[13] J N Mordeson and C-S Peng ldquoOperations on fuzzy graphsrdquoInformation Sciences vol 79 no 3-4 pp 159ndash170 1994

[14] M S Sunitha and A V Kumar ldquoComplement of a fuzzy graphrdquoIndian Journal of Pure and Applied Mathematics vol 33 no 9pp 1451ndash1464 2002

[15] H Ju and L Wang ldquoInterval-valued fuzzy subsemigroupsand subgroups associated by intervalvalued fuzzy graphsrdquo inProceedings of the WRI Global Congress on Intelligent Systems(GCIS rsquo09) pp 484ndash487 Xiamen China May 2009

[16] MAkram ldquoBipolar fuzzy graphsrdquo Information Sciences vol 181no 24 pp 5548ndash5564 2011

[17] M Akram ldquoBipolar fuzzy graphs with applicationsrdquo KnowledgeBased Systems vol 39 pp 1ndash8 2013

[18] M Akram ldquoInterval-valued fuzzy line graphsrdquoNeural Comput-ing and Applications vol 21 pp 145ndash150 2012

[19] M Akram and W A Dudek ldquoInterval-valued fuzzy graphsrdquoComputers amp Mathematics with Applications vol 61 no 2 pp289ndash299 2011

[20] M Akram and B Davvaz ldquoStrong intuitionistic fuzzy graphsrdquoFilomat vol 26 no 1 pp 177ndash196 2012

[21] M Akram K H Dar and K P Shum ldquoInterval-valued (120572120573)-fuzzy K-algebrasrdquo Applied Soft Computing Journal vol 11 no 1pp 1213ndash1222 2011

[22] T AL-Hawary ldquoComplete fuzzy graphsrdquo International Journalof Mathematical Combinatorics vol 4 pp 26ndash34 2011

[23] P Bhattacharya ldquoSome remarks on fuzzy graphsrdquo PatternRecognition Letters vol 6 no 5 pp 297ndash302 1987

[24] S M Chen ldquoInterval-valued fuzzy hypergraph and fuzzypartitionrdquo IEEE Transactions on Systems Man and CyberneticsB vol 27 no 4 pp 725ndash733 1997

[25] G Deschrijver and C Cornelis ldquoRepresentability in interval-valued fuzzy set theoryrdquo International Journal of UncertaintyFuzziness and Knowledge-Based Systems vol 15 no 3 pp 345ndash361 2007

[26] A N Gani and S R Latha ldquoOn irregular fuzzy graphsrdquoAppliedMathematical Sciences vol 6 no 9ndash12 pp 517ndash523 2012

[27] X Ma J Zhan B Davvaz and Y B Jun ldquoSome kinds of (isin isinor119902)-interval-valued fuzzy ideals of BCI-algebrasrdquo InformationSciences vol 178 no 19 pp 3738ndash3754 2008

[28] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Studies in Fuzziness and SoftComputing PhysicaHeidelberg Germany 1998

[29] F Riaz and KM Ali ldquoApplications of graph theory in computersciencerdquo in Proceedings of the 3rd International Conferenceon Computational Intelligence Communication Systems andNetworks (CICSyN rsquo11) pp 142ndash145 Bali Indonesia July 2011

[30] M Behzad and G Chartrand ldquoNo graph is perfectrdquo TheAmerican Mathematical Monthly vol 74 pp 962ndash963 1967

[31] G Deschrijver and E E Kerre ldquoOn the relationship betweensome extensions of fuzzy set theoryrdquo Fuzzy Sets and Systemsvol 133 no 2 pp 227ndash235 2003

[32] A Shannon and K T Atanassov ldquoA first step to a theory ofthe intuitionistic fuzzy graphsrdquo in Proceeding of the FUBESTD Lakov Ed pp 59ndash61 Sofia Bulgaria 1994

[33] R Parvathi M G Karunambigai and K T Atanassov ldquoOper-ations on intuitionistic fuzzy graphsrdquo in Proceedings of theIEEE International Conference on Fuzzy Systems pp 1396ndash1401August 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 7

1 2

4 3

[03 06] [04 05]

[04 05] [02 07]

[01 02]

[01 02]

[01 04] [01 04]

Figure 5 119866 is neighbourly totally irregular

Example 33 Consider an interval-valued fuzzy graph119866 suchthat

119881 = V1 V2 V3 V4

119864 = V1V2 V2V3 V3V4 V4V1

(27)

By routine computations we have deg[V1] = [11 16]

deg[V2] = [09 18] deg[V

3] = [10 17] and deg[V

4] =

[09 18] Hence119866 as shown in Figure 5 is neighbourly totallyirregular

Definition 34 Let 119866 be a connected interval-valued fuzzygraph 119866 is called highly irregular if every vertex of 119866 isadjacent to vertices with distinct neighbourhood degrees

Example 35 Consider an interval-valued fuzzy graph119866 suchthat

119881 = V1 V2 V3 V4 V5 V6

119864 = V1V2 V2V3 V2V6 V3V4 V3V5 V4V5 V5V1

(28)

By routine computations we have deg(V1) = [04 08]

deg(V2) = [06 17] deg(V

3) = [09 13] deg(V

4) = [06 11]

deg(V5) = [10 18] and deg(V

6) = [01 04] Clearly 119866 as

shown in Figure 6 is highly irregular

Example 36 Consider an interval-valued fuzzy graph119866 suchthat

119881 = V1 V2 V3 V4

119864 = V1V2 V2V3 V3V4 V4V1

(29)

By routine computations we have deg(V1) = [06 10]

deg(V2) = [08 09] deg(V

3) = [06 10] and deg(V

4) =

1 2

4

35

[02 02]

[01 03]

[03 04]

[02

04

][02

04]

[05 05]

[02 06] [01 04]

[01 04]

[03 07]

[01 0

4]

[02 02]

[02 03]

6

Figure 6 119866 is highly irregular

[08 09]We see that every two adjacent vertices have distinctopen neighbourhood degree But the vertex V

2adjacent to

the vertices V1and V

3has the same neighbourhood degree

that is deg(V1) = deg(V

3) Hence 119866 as shown in Figure 7 is

neighbourly irregular but not highly irregular

Remark 37 A neighbourly irregular interval-valued fuzzygraph may not be highly irregular

Theorem38 An interval-valued fuzzy graph119866 is highly irreg-ular and neighbourly irregular interval-valued fuzzy graph ifand only if the neighbourhood degrees of all the vertices of 119866are distinct

Proof Let 119866 be an interval-valued fuzzy graph with 119899-vertices V

1 V2 V

119899 Assume that 119866 is highly irregular and

neighbourly irregular

Claim 1 The neighbourhood degrees of all vertices of 119866

are distinct Let deg(V119894) = [119896

119894 119897119894] 119894 = 1 2 119899 Let

the adjacent vertices of V1be V2 V3 V

119899with neighbour-

hood degrees [1198962 1198972] [1198963 1198973] [119896

119899 119897119899] respectively Then

1198962

= 1198963

= sdot sdot sdot = 119896119899

and 1198972

= 1198973

= sdot sdot sdot = 119897119899 since 119866 is highly

irregular Also 1198961

= 1198962

= 1198963

= sdot sdot sdot = 119896119899and 1198971

= 1198972

= 1198973

= sdot sdot sdot = 119897119899

since 119866 is neighbourly irregular Hence the neighbourhooddegree of all the vertices of 119866 is distinct

8 Journal of Applied Mathematics

1 2

34

[03 04]

[01 04]

[04 05]

[01 02]

[01 02]

[02 05]

[05 05]

[01 04]

Figure 7 119866 is neighbourly irregular but not highly irregular

Conversely assume that the neighbourhood degrees of allthe vertices of 119866 are distinct

Claim 2 119866 is highly irregular and neighbourly irregularinterval-valued fuzzy graph

Let deg(V119894) = [119896

119894 119897119894] 119894 = 1 2 119899 Given that

1198961

= 1198962

= 1198963

= sdot sdot sdot = 119896119899and 1198971

= 1198972

= 1198973

= sdot sdot sdot = 119897119899 which implies

that every two adjacent vertices have distinct neighbourhooddegrees and to every vertex the adjacent vertices have distinctneighbourhood degrees

Theorem 39 An interval-valued fuzzy graph 119866 of 119866lowast where119866lowast is a cycle with 3 vertices that is neighbourly irregular andhighly irregular if and only if the lower and upper membershipvalues of the vertices between every pair of vertices are alldistinct

Proof Assume that lower and upper membership values ofthe vertices are all distinct

Claim 1 119866 is neighbourly irregular and highly irregularinterval-valued fuzzy graph

Let V119894 V119895 V119896

isin 119881 Given that 120583minus119860(V119894) = 120583minus119860(V119895) =

120583minus

119860(V119896) and 120583

+

119860(V119894) = 120583+

119860(V119895) = 120583+

119860(V119896) which implies that

sum119909isin119873(119909)

120583minus119860(V119894) = sum119909isin119873(119909)

120583minus119860(V119895) = sum119909isin119873(119909)

120583minus119860(V119896) and

sum119909isin119873(119909)

120583+119860(V119894) = sum119909isin119873(119909)

120583+119860(V119895) = sum119909isin119873(119909)

120583+119860(V119896) That is

deg(V119894) = deg(V

119895) = deg(V

119896) Hence 119866 is neighbourly

irregular and highly irregularConversely assume that 119866 is neighbourly irregular and

highly irregular

Claim 2 Lower and upper membership values of the verticesare all distinct

Let deg(V119894) = [119896

119894 119897119894] 119894 = 1 2 119899 Suppose that lower

and upper membership value of any two vertices are the

1

23

[04 06]

[02 06]

[02 07] [02 06]

[04 06]

[04 06]

Figure 8 119866 is not neighbourly irregular but it is complete

same Let V1 V2

isin 119881 Let 120583minus119860(V1) = 120583minus

119860(V2) and 120583+

119860(V1) =

120583+119860(V2) Then deg(V

1) = deg(V

2) since 119866lowast is cycle which is

a contradiction to the fact that119866 is neighbourly irregular andhighly irregular interval-valued fuzzy graph Hence lowermembership and upper membership value of the vertices areall distinct

Remark 40 A complete interval-valued fuzzy graph may notbe neighbourly irregular

Example 41 Consider an interval-valued fuzzy graph 119866 suchthat

119881 = V1 V2 V3 119864 = V

1V2 V2V3 V1V3 (30)

By routine computations we have deg(V1) = [06 13]

deg(V2) = [06 13] and deg(V

3) = [08 12] We see that

neighbourhood degree of V1and V

2is not distinct Hence

119866 as shown in Figure 8 is not neighbourly irregular but it iscomplete

Remark 42 A neighbourly total irregular interval-valuedfuzzy graph may not be neighbourly irregular

Example 43 Consider an interval-valued fuzzy graph119866 suchthat

119881 = V1 V2 V3 V4 119864 = V

1V2 V2V3 V3V4 V4V1 (31)

By routine computations we have deg(V1) = [07 09]

deg(V2) = [08 11] deg(V

3) = [07 09] deg(V

4) = [08 11]

deg[V1] = [11 15] deg[V

2] = [11 15] deg[V

3] = [11 14]

and deg[V4] = [12 16] We see that deg[V

1] = deg[V

2]

Hence 119866 as shown in Figure 9 is neighbourly irregular butnot a neighbourly total irregular

Journal of Applied Mathematics 9

1 2

34

[04 06]

[01 04]

[04 05] [01 02]

[01 02]

[04 05]

[03 04]

[01 04]

Figure 9 119866 is neighbourly irregular but not neighbourly totalirregular

Proposition 44 If an interval-valued fuzzy graph 119866 is neigh-bourly irregular and [120583minus

119860 120583+119860] is a constant function then it is

neighbourly totally irregular

Proof Assume that 119866 is a neighbourly irregular interval-valued fuzzy graph Then the open neighbourhood degreesof every two adjacent vertices are distinct Let V

119894 V119895

isin 119881 beadjacent vertices with distinct open neighbourhood degrees[1198961 1198971] and [119896

2 1198972] where 119896

1= 1198962 1198971

= 1198972 Let us assume that

(1205831(V119894) ]1(V119894)) = (120583

1(V119895) ]1(V119895)) = [119888

1 1198882] where 119888

1 1198882are

constant and 1198881 1198882isin [0 1] Therefore deg

120583minus

[V119894] = deg

120583minus

(V119894) +

1205831(V119894) = 119896

1+ 1198881 deg120583+

[V119894] = deg

120583+

(V119894) + ]1(V119894) = 119897

1+

1198882deg120583minus

[V119895] = deg

120583minus

(V119895) + 1205831(V119895) = 1198962+ 1198881 and deg

120583+

[V119895] =

deg120583+

(V119895) + ]1(V119895) = 1198972+ 1198882

Claim Consider that deg120583minus

[V119894] = deg

120583minus

[V119895] and deg

120583+

[V119894] =

deg120583+

[V119895] Suppose that deg

120583minus

[V119894] = deg

120583minus

[V119895] and

deg120583+

[V119894] = deg

120583+

[V119895] Consider that

deg120583minus

[V119894] = deg

120583minus

[V119895]

1198961+ 1198881= 1198962+ 1198881

1198961minus 1198962= 1198881minus 1198881= 0

1198961= 1198962 which is a contradiction to 119896

1= 1198962

(32)

Therefore deg120583minus

[V119894] = deg

120583minus

[V119895] Similarly we consider that

deg120583+

[V119894] = deg

120583+

[V119895]

1198971+ 1198882= 1198972+ 1198882

1198971minus 1198972= 1198882minus 1198882= 0

1198971= 1198972 which is a contradiction to 119897

1= 1198972

(33)

Therefore deg120583+

[V119894] = deg

120583+

[V119895] Hence 119866 is a neighbourly

totally irregular interval-valued fuzzy graph

Theorem 45 If an interval-valued fuzzy graph 119866 is neigh-bourly totally irregular and [120583minus

119860 120583+119860] is a constant function

then it is a neighbourly irregular interval-valued fuzzy graph

Proof Assume that119866 is a neighbourly total irregular interval-valued fuzzy graph Then the closed neighbourhood degreeof every two adjacent vertices is distinct Let V

119894 V119895

isin 119881

and deg[V119894] = [119896

1 1198971] deg[V

119895] = [119896

2 1198972] where 119896

1= 1198962

and 1198971

= 1198972 Assume that (120583

1(V119894) ]1(V119894)) = [119888

1 1198882] and

(1205831(V119895) ]1(V119895)) = [119888

1 1198882] where 119888

1 1198882

isin [0 1] are constantand deg[V

119894] = deg[V

119895]

Claim Consider that deg(V119894) = deg(V

119895)

Given that deg[V119894] = deg[V

119895] which implies deg

120583minus

[V119894] =

deg120583minus

[V119895] and deg

120583+

[V119894] = deg

120583+

[V119895] now we consider that

deg120583minus

[V119894] = deg

120583minus

[V119895]

1198961+ 1198881

= 1198962+ 1198881

1198961

= 1198962

(34)

We now consider that

deg120583+

[V119894] = deg

120583+

[V119895]

1198971+ 1198882

= 1198972+ 1198882

1198971

= 1198972

(35)

That is the neighbourhood degrees of adjacent vertices of 119866are distinct Hence neighbourhood degree of every pair ofadjacent vertices is distinct in 119866

Proposition 46 If an interval-valued fuzzy graph 119866 is neigh-bourly irregular and neighbourly totally irregular then [120583

minus

119860 120583+

119860]

need not be a constant function

Remark 47 If 119866 is a neighbourly irregular interval-valuedfuzzy graph then interval-valued subgraph 119867 = (119860

1015840 1198611015840) of119866 may not be neighbourly irregular

Remark 48 If 119866 is a totally irregular interval-valued fuzzygraph then interval-valued fuzzy subgraph 119867 = (1198601015840 1198611015840) of119866 may not be totally irregular

5 Relationship between IFGs and IVFGs

In 2003 Deschrijver and Kerre [31] established the rela-tionships between some extensions of fuzzy sets In thissection we present the relationship between extensions offuzzy graphs Shannon and Atanassov [32] introduced thenotion of an intuitionistic fuzzy graph Some operations onintuitionistic fuzzy graphs are discussed in [33]

Definition 49 (see [32]) By an intuitionistic fuzzy graph (IFGfor short) 119866 of a graph 119866

lowast we mean a pair 119866 = (119860 119861)

10 Journal of Applied Mathematics

Fuzzy sets

Interval-valued

fuzzy sets

Intuitionisticfuzzy sets

Interval-valued

fuzzy graphs

Intuitionisticfuzzy graphs

Figure 10 Links between models

Interval-valued

fuzzy graphs

Intuitionistic

fuzzy graphs

Fuzzy graphs

Figure 11 Link between IFGs and IVFGs

where 119860 = (120583119860 ]119860) is an intuitionistic fuzzy set on 119881 and

119861 = (120583119861 ]119861) is an intuitionistic fuzzy relation on 119864 such that

120583119861(119909119910) le min (120583

119860(119909) 120583

119860(119910))

]119861(119909119910) le max (]

119860(119909) ]

119860(119910))

(36)

for all 119909119910 isin 119864 The class of all IFGs on 119866lowast will be denoted byIFG(119866lowast)

Theorem 50 (see [33]) If 1198661and 119866

2are intuitionistic fuzzy

graphs then 1198661cap 1198662 1198661cup 1198662 and 119866

1are intuitionistic fuzzy

graphs

Ju and Wang introduced the notion of interval-valuedfuzzy graph (IVFG for short) in [15] Some operations oninterval-valued fuzzy graphs are discussed in [18] The classof all IVFGs on 119866

lowast will be denoted byIVFG(119866lowast)

Theorem 51 (see [19]) If 1198661and 119866

2are interval-valued fuzzy

graphs then1198661cap11986621198661cup1198662 and119866

1are interval-valued fuzzy

graphs

Lemma 52 (IVFG(119866lowast) cup cap) and (IFG(119866lowast) cup cap) arecomplete lattices

Theorem 53 The mapping

1205951 IVFG (119866

lowast) 997888rarr IFG (119866

lowast) (37)

defined by

1205951(119861) = ((119909 120583

minus

119860(119909) 1 minus 120583

+

119860(119909)) | 119909 isin 119881

(119909119910 120583minus

119861(119909119910) 1 minus 120583

+

119861(119909119910)) | 119909119910 isin 119864)

(38)

where119861 = ((119909 120583

minus

119860(119909) 120583

+

119860(119909)) | 119909 isin 119881

(119909119910 120583minus

119861(119909119910) 120583

+

119861(119909119910)) | 119909119910 isin 119864)

(39)

is an isomorphism between lattices (IVFG(119866lowast) cup cap) and(IFG(119866lowast) cup cap)

Remark 54 From a pure mathematical point of viewTheorem 53 shows that the two concepts intuitionistic fuzzygraphs and interval-valued fuzzy graphs are equivalent

In Figures 10 and 11 we present the relationships that existbetween different models In these figures a double arrowbetween two theories means that they are equivalent a singlearrow 119883 rarr 119884 denotes that 119884 is an extension of 119883 InFigure 10 a dash arrow 119883[119884 denotes that model 119884 is basedon the previous model 119883

6 Conclusions

Graph theory has several interesting applications in systemanalysis operations research computer applications andeconomics Since most of the time the aspects of graphproblems are uncertain it is nice to deal with these aspectsvia the methods of fuzzy systems It is known that fuzzygraph theory has numerous applications in modern scienceand engineering especially in the field of information theoryneural networks expert systems cluster analysis medicaldiagnosis traffic engineering network routing town plan-ning and control theory Since interval-valued fuzzy settheory is an increasingly popular extension of fuzzy settheory where traditional [0 1]-valued membership degreesare replaced by intervals in [0 1] that approximate the(unknown) membership degrees specific types of interval-valued fuzzy graphs have been introduced and investigatedin this paper The natural extension of this research work isthe application of interval-valued fuzzy graphs in the area ofsoft computing including neural networks expert systemsdatabase theory and geographical information systems

Acknowledgments

The authors are thankful to the referees for their valuablecomments The authors are also thankful to Professor SyedMansoor Sarwar and Dr Faisal Aslam for their valuablesuggestions

References

[1] S G Shirinivas S Vetrivel and N M Elango ldquoApplications ofgraph theory in computer sciencemdashan overviewrdquo InternationalJournal of Engineering Science and Technology vol 2 no 9 pp4610ndash4621 2010

Journal of Applied Mathematics 11

[2] N Deo Graph Theory with Applications to Engineering andComputer Science Prentice Hall Englewood Cliffs NJ USA1990

[3] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoningmdashIrdquo Information Sciencesvol 8 pp 199ndash249 1975

[4] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[5] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 pp338ndash353 1965

[6] J M Mendel Uncertain Rule-Based Fuzzy Logic Systems Intro-duction and New Directions Prentice-Hall Upper Saddle RiverNJ USA 2001

[7] M B Gorzałczany ldquoA method of inference in approximatereasoning based on interval-valued fuzzy setsrdquo Fuzzy Sets andSystems vol 21 no 1 pp 1ndash17 1987

[8] M B Gorzałczany ldquoAn interval-valued fuzzy inferencemethodmdashsome basic propertiesrdquo Fuzzy Sets and Systems vol31 no 2 pp 243ndash251 1989

[9] M K Roy and R Biswas ldquoI-V fuzzy relations and Sanchezrsquosapproach for medical diagnosisrdquo Fuzzy Sets and Systems vol47 no 1 pp 35ndash38 1992

[10] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986

[11] A Kauffman Introduction a la Theorie des Sous-emsemblesFlous vol 1 Masson et Cie 1973

[12] A Rosenfeld ldquoFuzzy graphsrdquo in Fuzzy Sets and Their Applica-tions L A Zadeh K S Fu and M Shimura Eds pp 77ndash95Academic Press New York NY USA 1975

[13] J N Mordeson and C-S Peng ldquoOperations on fuzzy graphsrdquoInformation Sciences vol 79 no 3-4 pp 159ndash170 1994

[14] M S Sunitha and A V Kumar ldquoComplement of a fuzzy graphrdquoIndian Journal of Pure and Applied Mathematics vol 33 no 9pp 1451ndash1464 2002

[15] H Ju and L Wang ldquoInterval-valued fuzzy subsemigroupsand subgroups associated by intervalvalued fuzzy graphsrdquo inProceedings of the WRI Global Congress on Intelligent Systems(GCIS rsquo09) pp 484ndash487 Xiamen China May 2009

[16] MAkram ldquoBipolar fuzzy graphsrdquo Information Sciences vol 181no 24 pp 5548ndash5564 2011

[17] M Akram ldquoBipolar fuzzy graphs with applicationsrdquo KnowledgeBased Systems vol 39 pp 1ndash8 2013

[18] M Akram ldquoInterval-valued fuzzy line graphsrdquoNeural Comput-ing and Applications vol 21 pp 145ndash150 2012

[19] M Akram and W A Dudek ldquoInterval-valued fuzzy graphsrdquoComputers amp Mathematics with Applications vol 61 no 2 pp289ndash299 2011

[20] M Akram and B Davvaz ldquoStrong intuitionistic fuzzy graphsrdquoFilomat vol 26 no 1 pp 177ndash196 2012

[21] M Akram K H Dar and K P Shum ldquoInterval-valued (120572120573)-fuzzy K-algebrasrdquo Applied Soft Computing Journal vol 11 no 1pp 1213ndash1222 2011

[22] T AL-Hawary ldquoComplete fuzzy graphsrdquo International Journalof Mathematical Combinatorics vol 4 pp 26ndash34 2011

[23] P Bhattacharya ldquoSome remarks on fuzzy graphsrdquo PatternRecognition Letters vol 6 no 5 pp 297ndash302 1987

[24] S M Chen ldquoInterval-valued fuzzy hypergraph and fuzzypartitionrdquo IEEE Transactions on Systems Man and CyberneticsB vol 27 no 4 pp 725ndash733 1997

[25] G Deschrijver and C Cornelis ldquoRepresentability in interval-valued fuzzy set theoryrdquo International Journal of UncertaintyFuzziness and Knowledge-Based Systems vol 15 no 3 pp 345ndash361 2007

[26] A N Gani and S R Latha ldquoOn irregular fuzzy graphsrdquoAppliedMathematical Sciences vol 6 no 9ndash12 pp 517ndash523 2012

[27] X Ma J Zhan B Davvaz and Y B Jun ldquoSome kinds of (isin isinor119902)-interval-valued fuzzy ideals of BCI-algebrasrdquo InformationSciences vol 178 no 19 pp 3738ndash3754 2008

[28] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Studies in Fuzziness and SoftComputing PhysicaHeidelberg Germany 1998

[29] F Riaz and KM Ali ldquoApplications of graph theory in computersciencerdquo in Proceedings of the 3rd International Conferenceon Computational Intelligence Communication Systems andNetworks (CICSyN rsquo11) pp 142ndash145 Bali Indonesia July 2011

[30] M Behzad and G Chartrand ldquoNo graph is perfectrdquo TheAmerican Mathematical Monthly vol 74 pp 962ndash963 1967

[31] G Deschrijver and E E Kerre ldquoOn the relationship betweensome extensions of fuzzy set theoryrdquo Fuzzy Sets and Systemsvol 133 no 2 pp 227ndash235 2003

[32] A Shannon and K T Atanassov ldquoA first step to a theory ofthe intuitionistic fuzzy graphsrdquo in Proceeding of the FUBESTD Lakov Ed pp 59ndash61 Sofia Bulgaria 1994

[33] R Parvathi M G Karunambigai and K T Atanassov ldquoOper-ations on intuitionistic fuzzy graphsrdquo in Proceedings of theIEEE International Conference on Fuzzy Systems pp 1396ndash1401August 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Applied Mathematics

1 2

34

[03 04]

[01 04]

[04 05]

[01 02]

[01 02]

[02 05]

[05 05]

[01 04]

Figure 7 119866 is neighbourly irregular but not highly irregular

Conversely assume that the neighbourhood degrees of allthe vertices of 119866 are distinct

Claim 2 119866 is highly irregular and neighbourly irregularinterval-valued fuzzy graph

Let deg(V119894) = [119896

119894 119897119894] 119894 = 1 2 119899 Given that

1198961

= 1198962

= 1198963

= sdot sdot sdot = 119896119899and 1198971

= 1198972

= 1198973

= sdot sdot sdot = 119897119899 which implies

that every two adjacent vertices have distinct neighbourhooddegrees and to every vertex the adjacent vertices have distinctneighbourhood degrees

Theorem 39 An interval-valued fuzzy graph 119866 of 119866lowast where119866lowast is a cycle with 3 vertices that is neighbourly irregular andhighly irregular if and only if the lower and upper membershipvalues of the vertices between every pair of vertices are alldistinct

Proof Assume that lower and upper membership values ofthe vertices are all distinct

Claim 1 119866 is neighbourly irregular and highly irregularinterval-valued fuzzy graph

Let V119894 V119895 V119896

isin 119881 Given that 120583minus119860(V119894) = 120583minus119860(V119895) =

120583minus

119860(V119896) and 120583

+

119860(V119894) = 120583+

119860(V119895) = 120583+

119860(V119896) which implies that

sum119909isin119873(119909)

120583minus119860(V119894) = sum119909isin119873(119909)

120583minus119860(V119895) = sum119909isin119873(119909)

120583minus119860(V119896) and

sum119909isin119873(119909)

120583+119860(V119894) = sum119909isin119873(119909)

120583+119860(V119895) = sum119909isin119873(119909)

120583+119860(V119896) That is

deg(V119894) = deg(V

119895) = deg(V

119896) Hence 119866 is neighbourly

irregular and highly irregularConversely assume that 119866 is neighbourly irregular and

highly irregular

Claim 2 Lower and upper membership values of the verticesare all distinct

Let deg(V119894) = [119896

119894 119897119894] 119894 = 1 2 119899 Suppose that lower

and upper membership value of any two vertices are the

1

23

[04 06]

[02 06]

[02 07] [02 06]

[04 06]

[04 06]

Figure 8 119866 is not neighbourly irregular but it is complete

same Let V1 V2

isin 119881 Let 120583minus119860(V1) = 120583minus

119860(V2) and 120583+

119860(V1) =

120583+119860(V2) Then deg(V

1) = deg(V

2) since 119866lowast is cycle which is

a contradiction to the fact that119866 is neighbourly irregular andhighly irregular interval-valued fuzzy graph Hence lowermembership and upper membership value of the vertices areall distinct

Remark 40 A complete interval-valued fuzzy graph may notbe neighbourly irregular

Example 41 Consider an interval-valued fuzzy graph 119866 suchthat

119881 = V1 V2 V3 119864 = V

1V2 V2V3 V1V3 (30)

By routine computations we have deg(V1) = [06 13]

deg(V2) = [06 13] and deg(V

3) = [08 12] We see that

neighbourhood degree of V1and V

2is not distinct Hence

119866 as shown in Figure 8 is not neighbourly irregular but it iscomplete

Remark 42 A neighbourly total irregular interval-valuedfuzzy graph may not be neighbourly irregular

Example 43 Consider an interval-valued fuzzy graph119866 suchthat

119881 = V1 V2 V3 V4 119864 = V

1V2 V2V3 V3V4 V4V1 (31)

By routine computations we have deg(V1) = [07 09]

deg(V2) = [08 11] deg(V

3) = [07 09] deg(V

4) = [08 11]

deg[V1] = [11 15] deg[V

2] = [11 15] deg[V

3] = [11 14]

and deg[V4] = [12 16] We see that deg[V

1] = deg[V

2]

Hence 119866 as shown in Figure 9 is neighbourly irregular butnot a neighbourly total irregular

Journal of Applied Mathematics 9

1 2

34

[04 06]

[01 04]

[04 05] [01 02]

[01 02]

[04 05]

[03 04]

[01 04]

Figure 9 119866 is neighbourly irregular but not neighbourly totalirregular

Proposition 44 If an interval-valued fuzzy graph 119866 is neigh-bourly irregular and [120583minus

119860 120583+119860] is a constant function then it is

neighbourly totally irregular

Proof Assume that 119866 is a neighbourly irregular interval-valued fuzzy graph Then the open neighbourhood degreesof every two adjacent vertices are distinct Let V

119894 V119895

isin 119881 beadjacent vertices with distinct open neighbourhood degrees[1198961 1198971] and [119896

2 1198972] where 119896

1= 1198962 1198971

= 1198972 Let us assume that

(1205831(V119894) ]1(V119894)) = (120583

1(V119895) ]1(V119895)) = [119888

1 1198882] where 119888

1 1198882are

constant and 1198881 1198882isin [0 1] Therefore deg

120583minus

[V119894] = deg

120583minus

(V119894) +

1205831(V119894) = 119896

1+ 1198881 deg120583+

[V119894] = deg

120583+

(V119894) + ]1(V119894) = 119897

1+

1198882deg120583minus

[V119895] = deg

120583minus

(V119895) + 1205831(V119895) = 1198962+ 1198881 and deg

120583+

[V119895] =

deg120583+

(V119895) + ]1(V119895) = 1198972+ 1198882

Claim Consider that deg120583minus

[V119894] = deg

120583minus

[V119895] and deg

120583+

[V119894] =

deg120583+

[V119895] Suppose that deg

120583minus

[V119894] = deg

120583minus

[V119895] and

deg120583+

[V119894] = deg

120583+

[V119895] Consider that

deg120583minus

[V119894] = deg

120583minus

[V119895]

1198961+ 1198881= 1198962+ 1198881

1198961minus 1198962= 1198881minus 1198881= 0

1198961= 1198962 which is a contradiction to 119896

1= 1198962

(32)

Therefore deg120583minus

[V119894] = deg

120583minus

[V119895] Similarly we consider that

deg120583+

[V119894] = deg

120583+

[V119895]

1198971+ 1198882= 1198972+ 1198882

1198971minus 1198972= 1198882minus 1198882= 0

1198971= 1198972 which is a contradiction to 119897

1= 1198972

(33)

Therefore deg120583+

[V119894] = deg

120583+

[V119895] Hence 119866 is a neighbourly

totally irregular interval-valued fuzzy graph

Theorem 45 If an interval-valued fuzzy graph 119866 is neigh-bourly totally irregular and [120583minus

119860 120583+119860] is a constant function

then it is a neighbourly irregular interval-valued fuzzy graph

Proof Assume that119866 is a neighbourly total irregular interval-valued fuzzy graph Then the closed neighbourhood degreeof every two adjacent vertices is distinct Let V

119894 V119895

isin 119881

and deg[V119894] = [119896

1 1198971] deg[V

119895] = [119896

2 1198972] where 119896

1= 1198962

and 1198971

= 1198972 Assume that (120583

1(V119894) ]1(V119894)) = [119888

1 1198882] and

(1205831(V119895) ]1(V119895)) = [119888

1 1198882] where 119888

1 1198882

isin [0 1] are constantand deg[V

119894] = deg[V

119895]

Claim Consider that deg(V119894) = deg(V

119895)

Given that deg[V119894] = deg[V

119895] which implies deg

120583minus

[V119894] =

deg120583minus

[V119895] and deg

120583+

[V119894] = deg

120583+

[V119895] now we consider that

deg120583minus

[V119894] = deg

120583minus

[V119895]

1198961+ 1198881

= 1198962+ 1198881

1198961

= 1198962

(34)

We now consider that

deg120583+

[V119894] = deg

120583+

[V119895]

1198971+ 1198882

= 1198972+ 1198882

1198971

= 1198972

(35)

That is the neighbourhood degrees of adjacent vertices of 119866are distinct Hence neighbourhood degree of every pair ofadjacent vertices is distinct in 119866

Proposition 46 If an interval-valued fuzzy graph 119866 is neigh-bourly irregular and neighbourly totally irregular then [120583

minus

119860 120583+

119860]

need not be a constant function

Remark 47 If 119866 is a neighbourly irregular interval-valuedfuzzy graph then interval-valued subgraph 119867 = (119860

1015840 1198611015840) of119866 may not be neighbourly irregular

Remark 48 If 119866 is a totally irregular interval-valued fuzzygraph then interval-valued fuzzy subgraph 119867 = (1198601015840 1198611015840) of119866 may not be totally irregular

5 Relationship between IFGs and IVFGs

In 2003 Deschrijver and Kerre [31] established the rela-tionships between some extensions of fuzzy sets In thissection we present the relationship between extensions offuzzy graphs Shannon and Atanassov [32] introduced thenotion of an intuitionistic fuzzy graph Some operations onintuitionistic fuzzy graphs are discussed in [33]

Definition 49 (see [32]) By an intuitionistic fuzzy graph (IFGfor short) 119866 of a graph 119866

lowast we mean a pair 119866 = (119860 119861)

10 Journal of Applied Mathematics

Fuzzy sets

Interval-valued

fuzzy sets

Intuitionisticfuzzy sets

Interval-valued

fuzzy graphs

Intuitionisticfuzzy graphs

Figure 10 Links between models

Interval-valued

fuzzy graphs

Intuitionistic

fuzzy graphs

Fuzzy graphs

Figure 11 Link between IFGs and IVFGs

where 119860 = (120583119860 ]119860) is an intuitionistic fuzzy set on 119881 and

119861 = (120583119861 ]119861) is an intuitionistic fuzzy relation on 119864 such that

120583119861(119909119910) le min (120583

119860(119909) 120583

119860(119910))

]119861(119909119910) le max (]

119860(119909) ]

119860(119910))

(36)

for all 119909119910 isin 119864 The class of all IFGs on 119866lowast will be denoted byIFG(119866lowast)

Theorem 50 (see [33]) If 1198661and 119866

2are intuitionistic fuzzy

graphs then 1198661cap 1198662 1198661cup 1198662 and 119866

1are intuitionistic fuzzy

graphs

Ju and Wang introduced the notion of interval-valuedfuzzy graph (IVFG for short) in [15] Some operations oninterval-valued fuzzy graphs are discussed in [18] The classof all IVFGs on 119866

lowast will be denoted byIVFG(119866lowast)

Theorem 51 (see [19]) If 1198661and 119866

2are interval-valued fuzzy

graphs then1198661cap11986621198661cup1198662 and119866

1are interval-valued fuzzy

graphs

Lemma 52 (IVFG(119866lowast) cup cap) and (IFG(119866lowast) cup cap) arecomplete lattices

Theorem 53 The mapping

1205951 IVFG (119866

lowast) 997888rarr IFG (119866

lowast) (37)

defined by

1205951(119861) = ((119909 120583

minus

119860(119909) 1 minus 120583

+

119860(119909)) | 119909 isin 119881

(119909119910 120583minus

119861(119909119910) 1 minus 120583

+

119861(119909119910)) | 119909119910 isin 119864)

(38)

where119861 = ((119909 120583

minus

119860(119909) 120583

+

119860(119909)) | 119909 isin 119881

(119909119910 120583minus

119861(119909119910) 120583

+

119861(119909119910)) | 119909119910 isin 119864)

(39)

is an isomorphism between lattices (IVFG(119866lowast) cup cap) and(IFG(119866lowast) cup cap)

Remark 54 From a pure mathematical point of viewTheorem 53 shows that the two concepts intuitionistic fuzzygraphs and interval-valued fuzzy graphs are equivalent

In Figures 10 and 11 we present the relationships that existbetween different models In these figures a double arrowbetween two theories means that they are equivalent a singlearrow 119883 rarr 119884 denotes that 119884 is an extension of 119883 InFigure 10 a dash arrow 119883[119884 denotes that model 119884 is basedon the previous model 119883

6 Conclusions

Graph theory has several interesting applications in systemanalysis operations research computer applications andeconomics Since most of the time the aspects of graphproblems are uncertain it is nice to deal with these aspectsvia the methods of fuzzy systems It is known that fuzzygraph theory has numerous applications in modern scienceand engineering especially in the field of information theoryneural networks expert systems cluster analysis medicaldiagnosis traffic engineering network routing town plan-ning and control theory Since interval-valued fuzzy settheory is an increasingly popular extension of fuzzy settheory where traditional [0 1]-valued membership degreesare replaced by intervals in [0 1] that approximate the(unknown) membership degrees specific types of interval-valued fuzzy graphs have been introduced and investigatedin this paper The natural extension of this research work isthe application of interval-valued fuzzy graphs in the area ofsoft computing including neural networks expert systemsdatabase theory and geographical information systems

Acknowledgments

The authors are thankful to the referees for their valuablecomments The authors are also thankful to Professor SyedMansoor Sarwar and Dr Faisal Aslam for their valuablesuggestions

References

[1] S G Shirinivas S Vetrivel and N M Elango ldquoApplications ofgraph theory in computer sciencemdashan overviewrdquo InternationalJournal of Engineering Science and Technology vol 2 no 9 pp4610ndash4621 2010

Journal of Applied Mathematics 11

[2] N Deo Graph Theory with Applications to Engineering andComputer Science Prentice Hall Englewood Cliffs NJ USA1990

[3] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoningmdashIrdquo Information Sciencesvol 8 pp 199ndash249 1975

[4] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[5] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 pp338ndash353 1965

[6] J M Mendel Uncertain Rule-Based Fuzzy Logic Systems Intro-duction and New Directions Prentice-Hall Upper Saddle RiverNJ USA 2001

[7] M B Gorzałczany ldquoA method of inference in approximatereasoning based on interval-valued fuzzy setsrdquo Fuzzy Sets andSystems vol 21 no 1 pp 1ndash17 1987

[8] M B Gorzałczany ldquoAn interval-valued fuzzy inferencemethodmdashsome basic propertiesrdquo Fuzzy Sets and Systems vol31 no 2 pp 243ndash251 1989

[9] M K Roy and R Biswas ldquoI-V fuzzy relations and Sanchezrsquosapproach for medical diagnosisrdquo Fuzzy Sets and Systems vol47 no 1 pp 35ndash38 1992

[10] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986

[11] A Kauffman Introduction a la Theorie des Sous-emsemblesFlous vol 1 Masson et Cie 1973

[12] A Rosenfeld ldquoFuzzy graphsrdquo in Fuzzy Sets and Their Applica-tions L A Zadeh K S Fu and M Shimura Eds pp 77ndash95Academic Press New York NY USA 1975

[13] J N Mordeson and C-S Peng ldquoOperations on fuzzy graphsrdquoInformation Sciences vol 79 no 3-4 pp 159ndash170 1994

[14] M S Sunitha and A V Kumar ldquoComplement of a fuzzy graphrdquoIndian Journal of Pure and Applied Mathematics vol 33 no 9pp 1451ndash1464 2002

[15] H Ju and L Wang ldquoInterval-valued fuzzy subsemigroupsand subgroups associated by intervalvalued fuzzy graphsrdquo inProceedings of the WRI Global Congress on Intelligent Systems(GCIS rsquo09) pp 484ndash487 Xiamen China May 2009

[16] MAkram ldquoBipolar fuzzy graphsrdquo Information Sciences vol 181no 24 pp 5548ndash5564 2011

[17] M Akram ldquoBipolar fuzzy graphs with applicationsrdquo KnowledgeBased Systems vol 39 pp 1ndash8 2013

[18] M Akram ldquoInterval-valued fuzzy line graphsrdquoNeural Comput-ing and Applications vol 21 pp 145ndash150 2012

[19] M Akram and W A Dudek ldquoInterval-valued fuzzy graphsrdquoComputers amp Mathematics with Applications vol 61 no 2 pp289ndash299 2011

[20] M Akram and B Davvaz ldquoStrong intuitionistic fuzzy graphsrdquoFilomat vol 26 no 1 pp 177ndash196 2012

[21] M Akram K H Dar and K P Shum ldquoInterval-valued (120572120573)-fuzzy K-algebrasrdquo Applied Soft Computing Journal vol 11 no 1pp 1213ndash1222 2011

[22] T AL-Hawary ldquoComplete fuzzy graphsrdquo International Journalof Mathematical Combinatorics vol 4 pp 26ndash34 2011

[23] P Bhattacharya ldquoSome remarks on fuzzy graphsrdquo PatternRecognition Letters vol 6 no 5 pp 297ndash302 1987

[24] S M Chen ldquoInterval-valued fuzzy hypergraph and fuzzypartitionrdquo IEEE Transactions on Systems Man and CyberneticsB vol 27 no 4 pp 725ndash733 1997

[25] G Deschrijver and C Cornelis ldquoRepresentability in interval-valued fuzzy set theoryrdquo International Journal of UncertaintyFuzziness and Knowledge-Based Systems vol 15 no 3 pp 345ndash361 2007

[26] A N Gani and S R Latha ldquoOn irregular fuzzy graphsrdquoAppliedMathematical Sciences vol 6 no 9ndash12 pp 517ndash523 2012

[27] X Ma J Zhan B Davvaz and Y B Jun ldquoSome kinds of (isin isinor119902)-interval-valued fuzzy ideals of BCI-algebrasrdquo InformationSciences vol 178 no 19 pp 3738ndash3754 2008

[28] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Studies in Fuzziness and SoftComputing PhysicaHeidelberg Germany 1998

[29] F Riaz and KM Ali ldquoApplications of graph theory in computersciencerdquo in Proceedings of the 3rd International Conferenceon Computational Intelligence Communication Systems andNetworks (CICSyN rsquo11) pp 142ndash145 Bali Indonesia July 2011

[30] M Behzad and G Chartrand ldquoNo graph is perfectrdquo TheAmerican Mathematical Monthly vol 74 pp 962ndash963 1967

[31] G Deschrijver and E E Kerre ldquoOn the relationship betweensome extensions of fuzzy set theoryrdquo Fuzzy Sets and Systemsvol 133 no 2 pp 227ndash235 2003

[32] A Shannon and K T Atanassov ldquoA first step to a theory ofthe intuitionistic fuzzy graphsrdquo in Proceeding of the FUBESTD Lakov Ed pp 59ndash61 Sofia Bulgaria 1994

[33] R Parvathi M G Karunambigai and K T Atanassov ldquoOper-ations on intuitionistic fuzzy graphsrdquo in Proceedings of theIEEE International Conference on Fuzzy Systems pp 1396ndash1401August 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 9

1 2

34

[04 06]

[01 04]

[04 05] [01 02]

[01 02]

[04 05]

[03 04]

[01 04]

Figure 9 119866 is neighbourly irregular but not neighbourly totalirregular

Proposition 44 If an interval-valued fuzzy graph 119866 is neigh-bourly irregular and [120583minus

119860 120583+119860] is a constant function then it is

neighbourly totally irregular

Proof Assume that 119866 is a neighbourly irregular interval-valued fuzzy graph Then the open neighbourhood degreesof every two adjacent vertices are distinct Let V

119894 V119895

isin 119881 beadjacent vertices with distinct open neighbourhood degrees[1198961 1198971] and [119896

2 1198972] where 119896

1= 1198962 1198971

= 1198972 Let us assume that

(1205831(V119894) ]1(V119894)) = (120583

1(V119895) ]1(V119895)) = [119888

1 1198882] where 119888

1 1198882are

constant and 1198881 1198882isin [0 1] Therefore deg

120583minus

[V119894] = deg

120583minus

(V119894) +

1205831(V119894) = 119896

1+ 1198881 deg120583+

[V119894] = deg

120583+

(V119894) + ]1(V119894) = 119897

1+

1198882deg120583minus

[V119895] = deg

120583minus

(V119895) + 1205831(V119895) = 1198962+ 1198881 and deg

120583+

[V119895] =

deg120583+

(V119895) + ]1(V119895) = 1198972+ 1198882

Claim Consider that deg120583minus

[V119894] = deg

120583minus

[V119895] and deg

120583+

[V119894] =

deg120583+

[V119895] Suppose that deg

120583minus

[V119894] = deg

120583minus

[V119895] and

deg120583+

[V119894] = deg

120583+

[V119895] Consider that

deg120583minus

[V119894] = deg

120583minus

[V119895]

1198961+ 1198881= 1198962+ 1198881

1198961minus 1198962= 1198881minus 1198881= 0

1198961= 1198962 which is a contradiction to 119896

1= 1198962

(32)

Therefore deg120583minus

[V119894] = deg

120583minus

[V119895] Similarly we consider that

deg120583+

[V119894] = deg

120583+

[V119895]

1198971+ 1198882= 1198972+ 1198882

1198971minus 1198972= 1198882minus 1198882= 0

1198971= 1198972 which is a contradiction to 119897

1= 1198972

(33)

Therefore deg120583+

[V119894] = deg

120583+

[V119895] Hence 119866 is a neighbourly

totally irregular interval-valued fuzzy graph

Theorem 45 If an interval-valued fuzzy graph 119866 is neigh-bourly totally irregular and [120583minus

119860 120583+119860] is a constant function

then it is a neighbourly irregular interval-valued fuzzy graph

Proof Assume that119866 is a neighbourly total irregular interval-valued fuzzy graph Then the closed neighbourhood degreeof every two adjacent vertices is distinct Let V

119894 V119895

isin 119881

and deg[V119894] = [119896

1 1198971] deg[V

119895] = [119896

2 1198972] where 119896

1= 1198962

and 1198971

= 1198972 Assume that (120583

1(V119894) ]1(V119894)) = [119888

1 1198882] and

(1205831(V119895) ]1(V119895)) = [119888

1 1198882] where 119888

1 1198882

isin [0 1] are constantand deg[V

119894] = deg[V

119895]

Claim Consider that deg(V119894) = deg(V

119895)

Given that deg[V119894] = deg[V

119895] which implies deg

120583minus

[V119894] =

deg120583minus

[V119895] and deg

120583+

[V119894] = deg

120583+

[V119895] now we consider that

deg120583minus

[V119894] = deg

120583minus

[V119895]

1198961+ 1198881

= 1198962+ 1198881

1198961

= 1198962

(34)

We now consider that

deg120583+

[V119894] = deg

120583+

[V119895]

1198971+ 1198882

= 1198972+ 1198882

1198971

= 1198972

(35)

That is the neighbourhood degrees of adjacent vertices of 119866are distinct Hence neighbourhood degree of every pair ofadjacent vertices is distinct in 119866

Proposition 46 If an interval-valued fuzzy graph 119866 is neigh-bourly irregular and neighbourly totally irregular then [120583

minus

119860 120583+

119860]

need not be a constant function

Remark 47 If 119866 is a neighbourly irregular interval-valuedfuzzy graph then interval-valued subgraph 119867 = (119860

1015840 1198611015840) of119866 may not be neighbourly irregular

Remark 48 If 119866 is a totally irregular interval-valued fuzzygraph then interval-valued fuzzy subgraph 119867 = (1198601015840 1198611015840) of119866 may not be totally irregular

5 Relationship between IFGs and IVFGs

In 2003 Deschrijver and Kerre [31] established the rela-tionships between some extensions of fuzzy sets In thissection we present the relationship between extensions offuzzy graphs Shannon and Atanassov [32] introduced thenotion of an intuitionistic fuzzy graph Some operations onintuitionistic fuzzy graphs are discussed in [33]

Definition 49 (see [32]) By an intuitionistic fuzzy graph (IFGfor short) 119866 of a graph 119866

lowast we mean a pair 119866 = (119860 119861)

10 Journal of Applied Mathematics

Fuzzy sets

Interval-valued

fuzzy sets

Intuitionisticfuzzy sets

Interval-valued

fuzzy graphs

Intuitionisticfuzzy graphs

Figure 10 Links between models

Interval-valued

fuzzy graphs

Intuitionistic

fuzzy graphs

Fuzzy graphs

Figure 11 Link between IFGs and IVFGs

where 119860 = (120583119860 ]119860) is an intuitionistic fuzzy set on 119881 and

119861 = (120583119861 ]119861) is an intuitionistic fuzzy relation on 119864 such that

120583119861(119909119910) le min (120583

119860(119909) 120583

119860(119910))

]119861(119909119910) le max (]

119860(119909) ]

119860(119910))

(36)

for all 119909119910 isin 119864 The class of all IFGs on 119866lowast will be denoted byIFG(119866lowast)

Theorem 50 (see [33]) If 1198661and 119866

2are intuitionistic fuzzy

graphs then 1198661cap 1198662 1198661cup 1198662 and 119866

1are intuitionistic fuzzy

graphs

Ju and Wang introduced the notion of interval-valuedfuzzy graph (IVFG for short) in [15] Some operations oninterval-valued fuzzy graphs are discussed in [18] The classof all IVFGs on 119866

lowast will be denoted byIVFG(119866lowast)

Theorem 51 (see [19]) If 1198661and 119866

2are interval-valued fuzzy

graphs then1198661cap11986621198661cup1198662 and119866

1are interval-valued fuzzy

graphs

Lemma 52 (IVFG(119866lowast) cup cap) and (IFG(119866lowast) cup cap) arecomplete lattices

Theorem 53 The mapping

1205951 IVFG (119866

lowast) 997888rarr IFG (119866

lowast) (37)

defined by

1205951(119861) = ((119909 120583

minus

119860(119909) 1 minus 120583

+

119860(119909)) | 119909 isin 119881

(119909119910 120583minus

119861(119909119910) 1 minus 120583

+

119861(119909119910)) | 119909119910 isin 119864)

(38)

where119861 = ((119909 120583

minus

119860(119909) 120583

+

119860(119909)) | 119909 isin 119881

(119909119910 120583minus

119861(119909119910) 120583

+

119861(119909119910)) | 119909119910 isin 119864)

(39)

is an isomorphism between lattices (IVFG(119866lowast) cup cap) and(IFG(119866lowast) cup cap)

Remark 54 From a pure mathematical point of viewTheorem 53 shows that the two concepts intuitionistic fuzzygraphs and interval-valued fuzzy graphs are equivalent

In Figures 10 and 11 we present the relationships that existbetween different models In these figures a double arrowbetween two theories means that they are equivalent a singlearrow 119883 rarr 119884 denotes that 119884 is an extension of 119883 InFigure 10 a dash arrow 119883[119884 denotes that model 119884 is basedon the previous model 119883

6 Conclusions

Graph theory has several interesting applications in systemanalysis operations research computer applications andeconomics Since most of the time the aspects of graphproblems are uncertain it is nice to deal with these aspectsvia the methods of fuzzy systems It is known that fuzzygraph theory has numerous applications in modern scienceand engineering especially in the field of information theoryneural networks expert systems cluster analysis medicaldiagnosis traffic engineering network routing town plan-ning and control theory Since interval-valued fuzzy settheory is an increasingly popular extension of fuzzy settheory where traditional [0 1]-valued membership degreesare replaced by intervals in [0 1] that approximate the(unknown) membership degrees specific types of interval-valued fuzzy graphs have been introduced and investigatedin this paper The natural extension of this research work isthe application of interval-valued fuzzy graphs in the area ofsoft computing including neural networks expert systemsdatabase theory and geographical information systems

Acknowledgments

The authors are thankful to the referees for their valuablecomments The authors are also thankful to Professor SyedMansoor Sarwar and Dr Faisal Aslam for their valuablesuggestions

References

[1] S G Shirinivas S Vetrivel and N M Elango ldquoApplications ofgraph theory in computer sciencemdashan overviewrdquo InternationalJournal of Engineering Science and Technology vol 2 no 9 pp4610ndash4621 2010

Journal of Applied Mathematics 11

[2] N Deo Graph Theory with Applications to Engineering andComputer Science Prentice Hall Englewood Cliffs NJ USA1990

[3] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoningmdashIrdquo Information Sciencesvol 8 pp 199ndash249 1975

[4] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[5] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 pp338ndash353 1965

[6] J M Mendel Uncertain Rule-Based Fuzzy Logic Systems Intro-duction and New Directions Prentice-Hall Upper Saddle RiverNJ USA 2001

[7] M B Gorzałczany ldquoA method of inference in approximatereasoning based on interval-valued fuzzy setsrdquo Fuzzy Sets andSystems vol 21 no 1 pp 1ndash17 1987

[8] M B Gorzałczany ldquoAn interval-valued fuzzy inferencemethodmdashsome basic propertiesrdquo Fuzzy Sets and Systems vol31 no 2 pp 243ndash251 1989

[9] M K Roy and R Biswas ldquoI-V fuzzy relations and Sanchezrsquosapproach for medical diagnosisrdquo Fuzzy Sets and Systems vol47 no 1 pp 35ndash38 1992

[10] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986

[11] A Kauffman Introduction a la Theorie des Sous-emsemblesFlous vol 1 Masson et Cie 1973

[12] A Rosenfeld ldquoFuzzy graphsrdquo in Fuzzy Sets and Their Applica-tions L A Zadeh K S Fu and M Shimura Eds pp 77ndash95Academic Press New York NY USA 1975

[13] J N Mordeson and C-S Peng ldquoOperations on fuzzy graphsrdquoInformation Sciences vol 79 no 3-4 pp 159ndash170 1994

[14] M S Sunitha and A V Kumar ldquoComplement of a fuzzy graphrdquoIndian Journal of Pure and Applied Mathematics vol 33 no 9pp 1451ndash1464 2002

[15] H Ju and L Wang ldquoInterval-valued fuzzy subsemigroupsand subgroups associated by intervalvalued fuzzy graphsrdquo inProceedings of the WRI Global Congress on Intelligent Systems(GCIS rsquo09) pp 484ndash487 Xiamen China May 2009

[16] MAkram ldquoBipolar fuzzy graphsrdquo Information Sciences vol 181no 24 pp 5548ndash5564 2011

[17] M Akram ldquoBipolar fuzzy graphs with applicationsrdquo KnowledgeBased Systems vol 39 pp 1ndash8 2013

[18] M Akram ldquoInterval-valued fuzzy line graphsrdquoNeural Comput-ing and Applications vol 21 pp 145ndash150 2012

[19] M Akram and W A Dudek ldquoInterval-valued fuzzy graphsrdquoComputers amp Mathematics with Applications vol 61 no 2 pp289ndash299 2011

[20] M Akram and B Davvaz ldquoStrong intuitionistic fuzzy graphsrdquoFilomat vol 26 no 1 pp 177ndash196 2012

[21] M Akram K H Dar and K P Shum ldquoInterval-valued (120572120573)-fuzzy K-algebrasrdquo Applied Soft Computing Journal vol 11 no 1pp 1213ndash1222 2011

[22] T AL-Hawary ldquoComplete fuzzy graphsrdquo International Journalof Mathematical Combinatorics vol 4 pp 26ndash34 2011

[23] P Bhattacharya ldquoSome remarks on fuzzy graphsrdquo PatternRecognition Letters vol 6 no 5 pp 297ndash302 1987

[24] S M Chen ldquoInterval-valued fuzzy hypergraph and fuzzypartitionrdquo IEEE Transactions on Systems Man and CyberneticsB vol 27 no 4 pp 725ndash733 1997

[25] G Deschrijver and C Cornelis ldquoRepresentability in interval-valued fuzzy set theoryrdquo International Journal of UncertaintyFuzziness and Knowledge-Based Systems vol 15 no 3 pp 345ndash361 2007

[26] A N Gani and S R Latha ldquoOn irregular fuzzy graphsrdquoAppliedMathematical Sciences vol 6 no 9ndash12 pp 517ndash523 2012

[27] X Ma J Zhan B Davvaz and Y B Jun ldquoSome kinds of (isin isinor119902)-interval-valued fuzzy ideals of BCI-algebrasrdquo InformationSciences vol 178 no 19 pp 3738ndash3754 2008

[28] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Studies in Fuzziness and SoftComputing PhysicaHeidelberg Germany 1998

[29] F Riaz and KM Ali ldquoApplications of graph theory in computersciencerdquo in Proceedings of the 3rd International Conferenceon Computational Intelligence Communication Systems andNetworks (CICSyN rsquo11) pp 142ndash145 Bali Indonesia July 2011

[30] M Behzad and G Chartrand ldquoNo graph is perfectrdquo TheAmerican Mathematical Monthly vol 74 pp 962ndash963 1967

[31] G Deschrijver and E E Kerre ldquoOn the relationship betweensome extensions of fuzzy set theoryrdquo Fuzzy Sets and Systemsvol 133 no 2 pp 227ndash235 2003

[32] A Shannon and K T Atanassov ldquoA first step to a theory ofthe intuitionistic fuzzy graphsrdquo in Proceeding of the FUBESTD Lakov Ed pp 59ndash61 Sofia Bulgaria 1994

[33] R Parvathi M G Karunambigai and K T Atanassov ldquoOper-ations on intuitionistic fuzzy graphsrdquo in Proceedings of theIEEE International Conference on Fuzzy Systems pp 1396ndash1401August 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Journal of Applied Mathematics

Fuzzy sets

Interval-valued

fuzzy sets

Intuitionisticfuzzy sets

Interval-valued

fuzzy graphs

Intuitionisticfuzzy graphs

Figure 10 Links between models

Interval-valued

fuzzy graphs

Intuitionistic

fuzzy graphs

Fuzzy graphs

Figure 11 Link between IFGs and IVFGs

where 119860 = (120583119860 ]119860) is an intuitionistic fuzzy set on 119881 and

119861 = (120583119861 ]119861) is an intuitionistic fuzzy relation on 119864 such that

120583119861(119909119910) le min (120583

119860(119909) 120583

119860(119910))

]119861(119909119910) le max (]

119860(119909) ]

119860(119910))

(36)

for all 119909119910 isin 119864 The class of all IFGs on 119866lowast will be denoted byIFG(119866lowast)

Theorem 50 (see [33]) If 1198661and 119866

2are intuitionistic fuzzy

graphs then 1198661cap 1198662 1198661cup 1198662 and 119866

1are intuitionistic fuzzy

graphs

Ju and Wang introduced the notion of interval-valuedfuzzy graph (IVFG for short) in [15] Some operations oninterval-valued fuzzy graphs are discussed in [18] The classof all IVFGs on 119866

lowast will be denoted byIVFG(119866lowast)

Theorem 51 (see [19]) If 1198661and 119866

2are interval-valued fuzzy

graphs then1198661cap11986621198661cup1198662 and119866

1are interval-valued fuzzy

graphs

Lemma 52 (IVFG(119866lowast) cup cap) and (IFG(119866lowast) cup cap) arecomplete lattices

Theorem 53 The mapping

1205951 IVFG (119866

lowast) 997888rarr IFG (119866

lowast) (37)

defined by

1205951(119861) = ((119909 120583

minus

119860(119909) 1 minus 120583

+

119860(119909)) | 119909 isin 119881

(119909119910 120583minus

119861(119909119910) 1 minus 120583

+

119861(119909119910)) | 119909119910 isin 119864)

(38)

where119861 = ((119909 120583

minus

119860(119909) 120583

+

119860(119909)) | 119909 isin 119881

(119909119910 120583minus

119861(119909119910) 120583

+

119861(119909119910)) | 119909119910 isin 119864)

(39)

is an isomorphism between lattices (IVFG(119866lowast) cup cap) and(IFG(119866lowast) cup cap)

Remark 54 From a pure mathematical point of viewTheorem 53 shows that the two concepts intuitionistic fuzzygraphs and interval-valued fuzzy graphs are equivalent

In Figures 10 and 11 we present the relationships that existbetween different models In these figures a double arrowbetween two theories means that they are equivalent a singlearrow 119883 rarr 119884 denotes that 119884 is an extension of 119883 InFigure 10 a dash arrow 119883[119884 denotes that model 119884 is basedon the previous model 119883

6 Conclusions

Graph theory has several interesting applications in systemanalysis operations research computer applications andeconomics Since most of the time the aspects of graphproblems are uncertain it is nice to deal with these aspectsvia the methods of fuzzy systems It is known that fuzzygraph theory has numerous applications in modern scienceand engineering especially in the field of information theoryneural networks expert systems cluster analysis medicaldiagnosis traffic engineering network routing town plan-ning and control theory Since interval-valued fuzzy settheory is an increasingly popular extension of fuzzy settheory where traditional [0 1]-valued membership degreesare replaced by intervals in [0 1] that approximate the(unknown) membership degrees specific types of interval-valued fuzzy graphs have been introduced and investigatedin this paper The natural extension of this research work isthe application of interval-valued fuzzy graphs in the area ofsoft computing including neural networks expert systemsdatabase theory and geographical information systems

Acknowledgments

The authors are thankful to the referees for their valuablecomments The authors are also thankful to Professor SyedMansoor Sarwar and Dr Faisal Aslam for their valuablesuggestions

References

[1] S G Shirinivas S Vetrivel and N M Elango ldquoApplications ofgraph theory in computer sciencemdashan overviewrdquo InternationalJournal of Engineering Science and Technology vol 2 no 9 pp4610ndash4621 2010

Journal of Applied Mathematics 11

[2] N Deo Graph Theory with Applications to Engineering andComputer Science Prentice Hall Englewood Cliffs NJ USA1990

[3] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoningmdashIrdquo Information Sciencesvol 8 pp 199ndash249 1975

[4] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[5] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 pp338ndash353 1965

[6] J M Mendel Uncertain Rule-Based Fuzzy Logic Systems Intro-duction and New Directions Prentice-Hall Upper Saddle RiverNJ USA 2001

[7] M B Gorzałczany ldquoA method of inference in approximatereasoning based on interval-valued fuzzy setsrdquo Fuzzy Sets andSystems vol 21 no 1 pp 1ndash17 1987

[8] M B Gorzałczany ldquoAn interval-valued fuzzy inferencemethodmdashsome basic propertiesrdquo Fuzzy Sets and Systems vol31 no 2 pp 243ndash251 1989

[9] M K Roy and R Biswas ldquoI-V fuzzy relations and Sanchezrsquosapproach for medical diagnosisrdquo Fuzzy Sets and Systems vol47 no 1 pp 35ndash38 1992

[10] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986

[11] A Kauffman Introduction a la Theorie des Sous-emsemblesFlous vol 1 Masson et Cie 1973

[12] A Rosenfeld ldquoFuzzy graphsrdquo in Fuzzy Sets and Their Applica-tions L A Zadeh K S Fu and M Shimura Eds pp 77ndash95Academic Press New York NY USA 1975

[13] J N Mordeson and C-S Peng ldquoOperations on fuzzy graphsrdquoInformation Sciences vol 79 no 3-4 pp 159ndash170 1994

[14] M S Sunitha and A V Kumar ldquoComplement of a fuzzy graphrdquoIndian Journal of Pure and Applied Mathematics vol 33 no 9pp 1451ndash1464 2002

[15] H Ju and L Wang ldquoInterval-valued fuzzy subsemigroupsand subgroups associated by intervalvalued fuzzy graphsrdquo inProceedings of the WRI Global Congress on Intelligent Systems(GCIS rsquo09) pp 484ndash487 Xiamen China May 2009

[16] MAkram ldquoBipolar fuzzy graphsrdquo Information Sciences vol 181no 24 pp 5548ndash5564 2011

[17] M Akram ldquoBipolar fuzzy graphs with applicationsrdquo KnowledgeBased Systems vol 39 pp 1ndash8 2013

[18] M Akram ldquoInterval-valued fuzzy line graphsrdquoNeural Comput-ing and Applications vol 21 pp 145ndash150 2012

[19] M Akram and W A Dudek ldquoInterval-valued fuzzy graphsrdquoComputers amp Mathematics with Applications vol 61 no 2 pp289ndash299 2011

[20] M Akram and B Davvaz ldquoStrong intuitionistic fuzzy graphsrdquoFilomat vol 26 no 1 pp 177ndash196 2012

[21] M Akram K H Dar and K P Shum ldquoInterval-valued (120572120573)-fuzzy K-algebrasrdquo Applied Soft Computing Journal vol 11 no 1pp 1213ndash1222 2011

[22] T AL-Hawary ldquoComplete fuzzy graphsrdquo International Journalof Mathematical Combinatorics vol 4 pp 26ndash34 2011

[23] P Bhattacharya ldquoSome remarks on fuzzy graphsrdquo PatternRecognition Letters vol 6 no 5 pp 297ndash302 1987

[24] S M Chen ldquoInterval-valued fuzzy hypergraph and fuzzypartitionrdquo IEEE Transactions on Systems Man and CyberneticsB vol 27 no 4 pp 725ndash733 1997

[25] G Deschrijver and C Cornelis ldquoRepresentability in interval-valued fuzzy set theoryrdquo International Journal of UncertaintyFuzziness and Knowledge-Based Systems vol 15 no 3 pp 345ndash361 2007

[26] A N Gani and S R Latha ldquoOn irregular fuzzy graphsrdquoAppliedMathematical Sciences vol 6 no 9ndash12 pp 517ndash523 2012

[27] X Ma J Zhan B Davvaz and Y B Jun ldquoSome kinds of (isin isinor119902)-interval-valued fuzzy ideals of BCI-algebrasrdquo InformationSciences vol 178 no 19 pp 3738ndash3754 2008

[28] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Studies in Fuzziness and SoftComputing PhysicaHeidelberg Germany 1998

[29] F Riaz and KM Ali ldquoApplications of graph theory in computersciencerdquo in Proceedings of the 3rd International Conferenceon Computational Intelligence Communication Systems andNetworks (CICSyN rsquo11) pp 142ndash145 Bali Indonesia July 2011

[30] M Behzad and G Chartrand ldquoNo graph is perfectrdquo TheAmerican Mathematical Monthly vol 74 pp 962ndash963 1967

[31] G Deschrijver and E E Kerre ldquoOn the relationship betweensome extensions of fuzzy set theoryrdquo Fuzzy Sets and Systemsvol 133 no 2 pp 227ndash235 2003

[32] A Shannon and K T Atanassov ldquoA first step to a theory ofthe intuitionistic fuzzy graphsrdquo in Proceeding of the FUBESTD Lakov Ed pp 59ndash61 Sofia Bulgaria 1994

[33] R Parvathi M G Karunambigai and K T Atanassov ldquoOper-ations on intuitionistic fuzzy graphsrdquo in Proceedings of theIEEE International Conference on Fuzzy Systems pp 1396ndash1401August 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 11

[2] N Deo Graph Theory with Applications to Engineering andComputer Science Prentice Hall Englewood Cliffs NJ USA1990

[3] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoningmdashIrdquo Information Sciencesvol 8 pp 199ndash249 1975

[4] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[5] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 pp338ndash353 1965

[6] J M Mendel Uncertain Rule-Based Fuzzy Logic Systems Intro-duction and New Directions Prentice-Hall Upper Saddle RiverNJ USA 2001

[7] M B Gorzałczany ldquoA method of inference in approximatereasoning based on interval-valued fuzzy setsrdquo Fuzzy Sets andSystems vol 21 no 1 pp 1ndash17 1987

[8] M B Gorzałczany ldquoAn interval-valued fuzzy inferencemethodmdashsome basic propertiesrdquo Fuzzy Sets and Systems vol31 no 2 pp 243ndash251 1989

[9] M K Roy and R Biswas ldquoI-V fuzzy relations and Sanchezrsquosapproach for medical diagnosisrdquo Fuzzy Sets and Systems vol47 no 1 pp 35ndash38 1992

[10] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986

[11] A Kauffman Introduction a la Theorie des Sous-emsemblesFlous vol 1 Masson et Cie 1973

[12] A Rosenfeld ldquoFuzzy graphsrdquo in Fuzzy Sets and Their Applica-tions L A Zadeh K S Fu and M Shimura Eds pp 77ndash95Academic Press New York NY USA 1975

[13] J N Mordeson and C-S Peng ldquoOperations on fuzzy graphsrdquoInformation Sciences vol 79 no 3-4 pp 159ndash170 1994

[14] M S Sunitha and A V Kumar ldquoComplement of a fuzzy graphrdquoIndian Journal of Pure and Applied Mathematics vol 33 no 9pp 1451ndash1464 2002

[15] H Ju and L Wang ldquoInterval-valued fuzzy subsemigroupsand subgroups associated by intervalvalued fuzzy graphsrdquo inProceedings of the WRI Global Congress on Intelligent Systems(GCIS rsquo09) pp 484ndash487 Xiamen China May 2009

[16] MAkram ldquoBipolar fuzzy graphsrdquo Information Sciences vol 181no 24 pp 5548ndash5564 2011

[17] M Akram ldquoBipolar fuzzy graphs with applicationsrdquo KnowledgeBased Systems vol 39 pp 1ndash8 2013

[18] M Akram ldquoInterval-valued fuzzy line graphsrdquoNeural Comput-ing and Applications vol 21 pp 145ndash150 2012

[19] M Akram and W A Dudek ldquoInterval-valued fuzzy graphsrdquoComputers amp Mathematics with Applications vol 61 no 2 pp289ndash299 2011

[20] M Akram and B Davvaz ldquoStrong intuitionistic fuzzy graphsrdquoFilomat vol 26 no 1 pp 177ndash196 2012

[21] M Akram K H Dar and K P Shum ldquoInterval-valued (120572120573)-fuzzy K-algebrasrdquo Applied Soft Computing Journal vol 11 no 1pp 1213ndash1222 2011

[22] T AL-Hawary ldquoComplete fuzzy graphsrdquo International Journalof Mathematical Combinatorics vol 4 pp 26ndash34 2011

[23] P Bhattacharya ldquoSome remarks on fuzzy graphsrdquo PatternRecognition Letters vol 6 no 5 pp 297ndash302 1987

[24] S M Chen ldquoInterval-valued fuzzy hypergraph and fuzzypartitionrdquo IEEE Transactions on Systems Man and CyberneticsB vol 27 no 4 pp 725ndash733 1997

[25] G Deschrijver and C Cornelis ldquoRepresentability in interval-valued fuzzy set theoryrdquo International Journal of UncertaintyFuzziness and Knowledge-Based Systems vol 15 no 3 pp 345ndash361 2007

[26] A N Gani and S R Latha ldquoOn irregular fuzzy graphsrdquoAppliedMathematical Sciences vol 6 no 9ndash12 pp 517ndash523 2012

[27] X Ma J Zhan B Davvaz and Y B Jun ldquoSome kinds of (isin isinor119902)-interval-valued fuzzy ideals of BCI-algebrasrdquo InformationSciences vol 178 no 19 pp 3738ndash3754 2008

[28] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Studies in Fuzziness and SoftComputing PhysicaHeidelberg Germany 1998

[29] F Riaz and KM Ali ldquoApplications of graph theory in computersciencerdquo in Proceedings of the 3rd International Conferenceon Computational Intelligence Communication Systems andNetworks (CICSyN rsquo11) pp 142ndash145 Bali Indonesia July 2011

[30] M Behzad and G Chartrand ldquoNo graph is perfectrdquo TheAmerican Mathematical Monthly vol 74 pp 962ndash963 1967

[31] G Deschrijver and E E Kerre ldquoOn the relationship betweensome extensions of fuzzy set theoryrdquo Fuzzy Sets and Systemsvol 133 no 2 pp 227ndash235 2003

[32] A Shannon and K T Atanassov ldquoA first step to a theory ofthe intuitionistic fuzzy graphsrdquo in Proceeding of the FUBESTD Lakov Ed pp 59ndash61 Sofia Bulgaria 1994

[33] R Parvathi M G Karunambigai and K T Atanassov ldquoOper-ations on intuitionistic fuzzy graphsrdquo in Proceedings of theIEEE International Conference on Fuzzy Systems pp 1396ndash1401August 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of