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REGULAR INTERVAL-VALUED INTUITIONISTIC FUZZY SOFT GRAPH
Dr.N.Sarala1
Research Supervisor, Associate Professor,
Department of Mathematics,
A.D.M College for Women (Autonomous), Nagapattinam,
Affiliated to Bharathidasan University, Thiruchirupalli,Tamilnadu, India
Saralaadmc68@gmail.com
R.Deepa2
Research Scholar (Part Time),
Department of Mathematics,
A.D.M College for Women (Autonomous), Nagapattinam,
Affiliated to Bharathidasan University, Thiruchirupalli, Tamilnadu, India
Srideepamuruga13@gmail.com
R.Deepa3
Associate Professor,
Department of Mathematics,
E.G.S Pillay Engineering College (Autonomous), Nagapattinam,
Tamilnadu, India
ABSTRACT
In this paper, we introduce regular interval-valued intuitionistic fuzzy soft graphs
and investigate some of their attributes. We talk about f-morphism on an interval-valued
intuitionistic fuzzy soft graph and regular interval-valued intuitionistic fuzzy soft graphs. (2, u)-
regular and totally (2, u) regular interval-valued intuitionistic fuzzy soft graphs.
KEYWORDS :Intutionistic fuzzy soft graph, f-morphism,(2,u) regular soft graph
1. INTRODUCTION
In 1965, zadeh [9] introduced the concept of fuzzy set as a method of finding uncertainty. In
1975, Rosenfeld [7] introduced the concept of fuzzy graphs. Yeh and Bang [8] also introduced
fuzzy graphs independently. Fuzzy graphs are useful to represent relationships which deal with
uncertainty and it differs greatly from classical graphs. It has numerous applications to
problems in computer science, electrical engineering, system analysis, operation research,
economics, networking routing, transportation, etc. interval-valued Fuzzy Graphs are defined
by Akram and Dudec in 2011. Atanassov [5] introduced the concept of intuitionistic fuzzy
relations and intuitionistic Fuzzy Graph. In fact interval-valued intuitionistic fuzzy graphs and
interval-valued intuitionistic fuzzy graphs are two different models that extend theory of fuzzy
graph S.N.Mishra and A.Pal [6] introduces the product of interval values intuitionistic fuzzy
graph.
2. PRILIMINARIES
We start this section by reviewing some fundamental concepts related to FSG.
Definition 2.1: A fuzzy set of a non-empty base set π = π₯1, π₯2, β¦ . . π₯π is defined by its degree
of membership function π ; where π: π βΆ 0,1 assigning to all π₯1 β π , the degree to
which π β π.
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Definition 2.2: A fuzzy graph πΊ = (π, πΈ) is defined as a pair of function π: π βΆ 0,1 and
π: π Γ π βΆ 0,1 , where πΈ π₯π , π₯π β€ π π₯π β§ π π₯π , β π₯π , π₯π β π Γ π, βπ₯π , π₯π β π. Here π and
πΈ are known as node and link of πΊ = (π, πΈ) correspondingly.
Definition 2.3: Consider πΊ1 = π₯1, πΈ1 and πΊ2 = π₯2, πΈ2 are fuzzy graphs over the given set π.
The union operation of πΊ1 and πΊ2 is provides a fuzzy graph πΊ3 = π₯3, πΈ3 over the set π. Here π₯3 = π₯1 β¨ π₯2 = πππ₯ π1 π₯π , π2 π₯π , βπ₯π β π,
And π = 1,2,3 β¦π . Similarly πΈ3 π₯π , π₯π = πππ₯ πΈ1 π₯π , π₯π , πΈ2 π₯π , π₯π , β π₯π , π₯π β π Γ π ,
whereπ, π = 1,2,3 β¦π.
Definition 2.4: [8] The FSG is defined by 4 tuple as πΊ = (πΊβ, πΉ1, πΉ2 , π) such that
1. πΊβ = π, πΈ is a simple graph,
2. π is a nonempty set of attributes,
3. πΉ1, π is a FSS over π,
4. (πΉ2, π) is a FSS over πΈ,
5. (πΉ1 π , πΉ2 π ) is a fuzzy soft graph of πΊβ , βπ β π . That
is, πΉ2 π β€ πππ πΉ1 π π1 , πΉ1 π π2 , βπ β π and π1, π2 β π. Note that πΉ2 π π1π2 =
0, βπ1π2 β π Γ π β πΈ and βπ β π. The fuzzy soft graph (πΉ1 π , πΉ2 π ) is defined by
π» π for simplicity.
Definition 2.5: [8] A fuzzy soft graph πΊ is a strong FSG if π» π is a strong fuzzy graph for
all π β π, That is, πΉ2 π (ππππ) = πππ πΉ1 π ππ , πΉ1 π ππ for all ππ ππ β πΈ.
Definition 2.6: [8] Let πΊπ = (πΊπβ, πΉ1π , πΉ2π , ππ) and πΊπ = (πΊπ
β, πΉ1π , πΉ2π , ππ) be two FSGs of πΊπβ
and πΊπβ
, correspondingly. The union of πΊπ and πΊπ , symbolized by πΊπ βͺ πΊπ , is a FSG
(πΉ1, πΉ2, ππ βͺ ππ), such that (πΉ1, ππ βͺ ππ), is a FSS over π = ππ βͺ ππ , (πΉ2, ππ βͺ ππ) is a FSS
over πΈπ βͺ πΈπ , and π» π = (πΉ1 π , πΉ2 π ) is a fuzzy soft graph for all π β ππ βͺ ππ given by
π» π = {π»π π , ππ π β ππ β ππ π»π π , ππ π β ππ β ππ
π»π π βͺ π»π π , ππ π β ππ β© ππ
Definition 2.7: [9]
Let πΊ1 = (G*,πΉ1
, πΎ1 , A) and πΊ2
= (G*,πΉ2 ,πΎ2
, A) be two fuzzy soft graphs. A homomorphism f :
πΊ1 β πΊ2
is a mapping f : V1β V2 which satisfies the following conditions.
(i) πΉ1 (a) (x) β€ πΉ2
(a)(f(x))
π²π (a)(xy) β€ π²π
(a) (f (x)f(y)) for all aβA, x, y β V1, x y β E
DEFINITION: 2.8
Let G = (G*, FΝ , KΝ , B) be a simple graph, Y is a non-empty set and it is defined as
Y= {y1, y2 β¦ yn}, E β YXY, P (set of attributes) and A β P.
Also consider i)D1 is a m deg given by D1: BβIs
bβ D1 (b) =D1b, b β B,
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x
D1b: πΎβ [0, 1], πΎ1βD1b (y1)
(B, D1) denote an intuitionistic fuzzy soft node of the m degree
T1: BβΞs
bβT1 (b) =T1b, bβB and T1b: Yβ [0.1]
yiβ T1b (yi)
(B, T1) denote an intuitionistic fuzzy soft node of the nm degree such that
0 β€ D1a (yi) + T1a (yi) β€ 1, β yi β Y and b β B
ii) D2 is a m degree given on E and given by
D2: BβΞs (yxy)
bβD2 (b) = D2b (b) = D2b, b β B and
D2b: yxyβ [0, 1]
(yi, yj)β D2b (yi, yj)
T2 is a nm deg and defined on E by
T2: BβΞs (yxy)
bβ T2 (b) = T2b, b β B
T2b: yxyβ [0, 1]
(yi, yj)β S2b (yi, yj)
Where (B, D2) and (B, T2) are I FSG links of m deg and nm deg satisfying
a) D2b (yi, yj) β€ min {D1b (yi), D1b (yj)}
b) T2b (yi, yj) β€ max {T1b (yi), T1b (yj)}and
c) 0 β€ D2b (yi, yj) + T2b (yi, yj) β€ 1
0 β€ D2b (yi, yj), T2b (yi, yj), F (yi, yj) β€ 1, β (yi ,yj) β X
The graph G = (G*, B, Y, E) = Y, E, (B, D1), (B, T1), (B, D2), (B, T2) is known as the
Intuitionistic fuzzy soft graph.
DEFINITION: 2.9
An internal-valued intuitionistic fuzzy soft graph G = (G*, FΝ , KΝ , (x, y)) is called strong
interval valued intuitionistic fuzzy soft graph
If D-y (ab) = min (D
-x (a), D
- (b)) and
T-y (ab) = min (T
-x (a), T
-x (b))
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D+
y (ab) = min (D+
x (a), D+
x (b)) and
T+
y (ab) = max (T+
x (a), T+
x (b)) β ab β E.
3. REGULAR INTERVAL-VALUED INTUITIONISTIC FUZZY SOFT GRAPH
DEFINITION: 3.1
An internal-valued intuitionistic fuzzy soft graph G is said to be regular if the absolute
degree of each vertex of an interval-valued intuitionistic fuzzy soft graph is constant. If the
absolute degree of each vertex is u, then we say the graph is u-regular interval valued
intuitionistic fuzzy soft graph.
DEFINITION: 3.2
Absolute degree d (u) of any vertex u of an internal-valued intuitionistic fuzzy soft graph
G is
d(u) = | βuβ v,vπV π·B+ (u,v) - β uβ v,vπV πB
+( u,v) |
Absolute membership of an edge e=uv β e β G is defined as
d(e) = | D+
B - T+
B | , where e β (D, T) β e β G.
Example:3.2
Let G* = (V,E) where v={a1,a2,a3,a4} and
E = {a1a2, a2a3, a3a4, a1a4, a2a4, a1a3}, parameter {e} show in figure1.
Define G (A, B) by
DA (e) = {a1|(0.4,0 .7), a2|(0.5, 0.8), a3|(0.4,0 .8), a4|(0.3,0 .6)}
DB (e) = {a1a2|(0.4, 0.6), a2a3|(0.3,0 .5), a3a4|(0.3,0 .5), a1a4|(0.3,0 .5), a2a4|(0.3,0 .6),
a1a3|(0.4, 0.7)}
TA (e) = {a1|(0.3, 0.5), a2|(0.2,0 .4), a3|(0.1, 0.3), a4|(0.4, 0.6)}
TB (e) = {a1a2|(0.3, 0.5), a2a3|(0.2,0 .4), a3a4|(0.4,0 .6), a1a4|(0.4,0 .6), a2a4|(0.4, 0.6)
, a1a3|(0.3, 0.5)} a1 a2
a4 a3
Figure 1.
Absolute degree of an internal-valued intuitionistic fuzzy soft graph
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B B
Now, absolute degree of the vertices a1, a2, a3, a4 are
d(a1) = |(0. 7 +0 .6 + 0.5) β 0.5 +0 .5 +0 .6)| = |1.8 β 1.6| =0 .2
d(a2) = |(0. 6 + 0.6 +0 .5) β 0.5 +0 .4 +0 .6)| = |1.7 β 1.5| =0 .2
d(a3) = |(0. 7 +0 .5 + 0.5) β 0.4 +0 .5 +0 .6)| = |1.7 β 1.5| = 0.2
d(a4) = |(0. 5 + 0.6 +0 .5) β 0.6 + 0.6 +0 .6)| = |1.6 β 1.8| = 0.2
Here absolute degree of each vertex is 0.2. Thus, internal-valued intuitionistic fuzzy soft graph
G is 2-regular.
Definition: 3.3
Let G = (G*, FΝ , KΝ , (A, B)) be an internal-valued intuitionistic fuzzy soft graph on
G*= (V,E). The total degree of a vertex a1 is defined as
t d(a1) = | βπ1β π2,π2ππ£ π·B+ (a1,a2) - βπ1β π2,π2ππ£ πB
+(a1,a2) | + | D+A (a1) β T+
A(a1) |
= d (a1) + | D+
A (a1) β T+
A (a1) | β a1a2 β E.
If each vertex of G has the same total degree u; then G is said to be totally regular interval-
valued intuitionistic fuzzy soft graph.
Definition:3.4
Let G = (G*, FΝ , KΝ , (A, B)) be an internal-valued intuitionistic fuzzy soft graph. The d2
degree of a vertex a1β G is d2 (a1) = | β D 2+
(a1, a2) - β T 2+
(a1, a2) | and summation runs over all
such a1β V which are distance two apart from a1.
Where
DB2+ (a1, a2) = inf {DB
+ (a1, a2), DB+ (a1, a2)}
and
TB2+ (a1, a2) = sup {TB
+ (a1, a2), TB+ (a1, a2)}
Also,
DB+ (a1, a2) = 0 and TB
+ (a1, a2) = 1, for a1 a2 β πΈ
The minimum d2-degree of G is πΏ2 (G) = β§ {d2 (a1): a1β V}.
The maximum d2-degree of G is β2 (G) = β{d2 (a2): a2 β V}.
Example:3.4
Consider G* = (V, E), where v= {a1, a2, a3, a4} and
E = {a1a2, a2a3, a3a4, a4a5, a5a1}. Define G = (G*, FΝ , KΝ , (A, B)) by
DA (e) = {a1|(0.4,0 .7), a2|(0.5,0 .8), a3|(0.4,0 .8), a4|(0. 3,0 .6), a5|(0.3,0 .7)}
DB (e) = {a1a2|(0.4,0 .6), a2a3|(0.3, 0.5), a3a4|(0.3, 0.5), a4a5|(0.3,0 .5), a5a1|(0.3,0 .7)}
TA (e) = {a1|(0.3, 0.5), a2|(0.2,0 .4), a3|(0.1,0 .3), a4|(0.4,0 .6), a5|(0.2,0 .4)}
TB (e) = {a1a2|(0.3,0 .5), a2a3|(0.2, 0.4), a3a4|(0.4, 0.7), a4a5|(0.4,0 .6), a5a1|(0.3,0 .5)}
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a1
a2 a5
a3 a4
Figure 2.
d2-degree for the vertices of an interval-valued intuitionistic fuzzy soft graph
Now,
d2 (a1) = | inf {0.6,0 .5} + inf {0.7, 0.5} β sup {0.5, 0.4} β sup {.5, .6} | =0 .1
d2 (a2) = | inf {0.5, 0.5} + inf {0.6,0 .7} β sup {0.4,0 .7} β sup {0.5, 0.5} | = 0.1
d2 (a3) = | inf {0.5, 0.5} + inf {0.5, 0.6} β sup {0.7,0 .6} β sup {0.4, 0.5} | =0 .2
d2 (a4) = | inf {0.5, 0.7} + inf {0.5, 0.5} β sup {0.6,0 .5} β sup {0.7, 0.4} | =0 .3
d2 (a5) = | inf {0.7,0 .6} + inf {0.5,0 .5} β sup {0.5, 0.5} β sup {0.6, 0.7} | =0 .1
Theorem 3.1
Even length interval-valued intuitionistic fuzzy cycle soft graph is regular or u-
regularβΊ absolute membership of e and d2 (e) for each e β G is equal i.e., d(e)=d2(e) β e β G.
Proof
Let G = (G*, FΝ , KΝ , (A, B)) is an even length interval-valued intuitionistic fuzzy cycle soft
graph then if the absolute membership of each edge is same i.e., equal to any real number u
then d(e)=d2(e) β e β G thus d(a1)=2u β a1 βG. Hence the theorem is trivially true. Now if the
absolute membership of any two adjacent edges is not equal but d2 (e) is equal then for any e β
G.
d (e1) = d2 (e2) = d (e3) =β¦β¦.= d (e2n-1) = u1 (say)
Similarly,
d (e2) = d2 (e2) = d (e4) = d2 (e4)β¦β¦.= d (e2n) = u2 (say)
Since cycle is of even length thus, there must be n number of eiβs having absolute
membership u1 and u2
Also, we know that for a cycle absolute degree of any vertex a1 is
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d(a1) = | βπ1β π2,π2ππ£ π·B+ (a1,a2) - βπ1β π2,π2ππ£ πB
+(a1,a2) | = d(ei) + d(ei+1) = u1 + u2
Therefore, d (a1) = u β a1β G so G is regular. Hence the theorem
Theorem 3.2
Cartesian product of two regular interval-valued intuitionistic fuzzy soft graph G1 and
G2 is regular if G1 is a weak regular interval-valued intuitionistic fuzzy soft graph sub graph of
G2 or vice versa.
Proof
Let G1 and G2 be two regular interval-valued intuitionistic fuzzy soft graph then the Cartesian
product of G1 and G2 is regular if the absolute membership of each arc e of G1ΓG2 is equal and
this is possible if d (e) = min {d(ei),d(ei)}, where ei β G1 and ej β G2 for all i and j thus the
condition is necessary for regularity of G1ΓG2 is either of G1 or G2 be a weak regular soft sub
graph of each other. Now, let G1 is weak regular sub graph G2 then we know that each edge of
G1ΓG2 get interval-valued membership and non-membership as minimum of D1 and D2 and
maximum of T1 and T2 thus, if G1 is weak then D1 and T1 dominates all the arc of G1ΓG2.So all
the arc receive same absolute membership which imply G1ΓG2 is regular. Hence the theorem.
Theorem 3.3
Any interval-valued intuitionistic fuzzy soft path graph of length l is never an regular interval-
valued intuitionistic fuzzy soft graph l >1.
Proof
For any interval-valued intuitionistic fuzzy soft path graph G = (G*, FΝ , KΝ , (A, B)) either
every edge have same absolute membership or some edges have district absolute membership.
Thus when all edges receive same absolute membership then at least both the end vertices of the
path soft graph G get different absolute degree then in-vertices of the soft path graph hence G is
not regular. Similarly if some edges have district absolute membership, let d(e1) β d(e2) and
both e1 and e2 are adjacent let a1 be the common vertex of e1 and e2βΉd(a1) is always greater than
other and vertices of e1 and e2 which imply G is not regular. For l=1 graph is always regular
because in this case absolute membership of an edge become the absolute degree of the vertices.
Hence the theorem.
3. (A). (2, u)-Regular and Totally (2, u) - Regular Interval-Valued Intuitionistic Fuzzy Soft
Graph
Definition: 3. (a) (1) Let G = (G*, FΝ, KΝ, (A, B)) be an interval-valued intuitionistic fuzzy soft
graph on G*(V, E). If d2(a2) = 2, βa2βV then G is said to be (2, u)-regular interval -valued
intuitionistic fuzzy soft graph.
Example: 3. (a) (1) Consider G*(V, E) where v= {a1, a2, a3, a4} and E = {a1a2, a2a3, a3a4, a4a1,}
and {e} be parameter set .Define G = (G*, FΝ, KΝ, (A, B)) by
DA (e) = {a1| (0.4, 0.7), a2| (0.5, 0 .8), a3| (0.4, 0 .8), a4| (0.3, 0.6)}
DB (e) = {a1a2| (0.4, 0.6), a2a3| (0.3, 0.5), a3a4| (0.3, 0 .6), a4a1| (0.3, 0.5)}
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TA (e) = {a1| (0.3, 0.5), a2| (0.2, 0.4), a3| (0.1, 0.3), a4| (0.3, 0 .5)}
TB (e) = {a1a2| (0.3, 0 .5), a2a3| (0.2, 0 .4), a3a4| (0.3, 0.5), a4a1| (0.4, 0 .7)}
Now,
a1 a2
a3 a4
Figure 3.
(2, u)-Regular interval-valued intuitionistic fuzzy soft graph
d2 (a1) = | inf {0.6, 0 .5} + inf {0.5, 0.6} β sup {0.5, 0 .4} β sup {0.7, 0 .5} | = 0.2,
d2 (a2) = | inf {0.5, 0.6} + inf {0.6, 0 .5} β sup {0.4, 0.5} β sup {0.5, 0.7} | =0.2,
d2 (a3) = | inf {0.6, 0.5} + inf {0.5, 0.6} β sup {0.5, 0 .7} β sup {0.4, 0 .5} | =0.2,
d2(a4) = | inf {0.5, 0.6} + inf {0.6,0 .5} β sup {0.7, 0.5} β sup {0.5,0 .4} | =0 .2,
Here d2(a1)=d2(a2)= d2(a3)= d2(a4)=0.2 thus the graph G is (2,2)-regular interval-
valued intuitionistic fuzzy soft graph.
Theorem: 3. (a) (1)
Let G = (G*, FΝ, KΝ, (A, B)) be a strong interval-valued intuitionistic fuzzy soft graph
on G*= (V, E) then D
+A (a1) =c1 and T
+A (a1) =c2 for all a1βV if and only if the following
conditions are equivalent.
i) G = (G*, FΝ, KΝ, (A, B)) is a (2, u)-regular interval-valued intuitionistic fuzzy soft graph.
β ±) G = (G*, FΝ, KΝ, (A, B)) is a totally (2, u+c) - regular interval-valued intuitionistic fuzzy soft graph where
c = | c1 βc2 |.
Proof
Let D+
A (a1) =c1 and T+
A (a1) =c2 for all a1βV.
Thus | D+
A (a1) - T+
A (a1) | = | c1 βc2 | = c for all a1βV. Suppose that G = (G*, FΝ , KΝ , (A, B)) is a
(2, u)- Regular interval-valued intuitionistic fuzzy soft graph then d2 (a1) =u, for all a1βV.
Hence,
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t d2 (a1) =d2 (a1) + | D+
A (a1) - T+A (a1) | βΉ t d2 (a1) = u+c, β a1 β V.
Hence, G = (G*, FΝ, KΝ, (A, B)) is a totally (2, u+c))-regular interval-valued intuitionistic fuzzy
soft graph.
Thus (i) βΉ (ii) is proved.
Suppose, G = (G*, FΝ, KΝ, (A, B)) is a totally (2, u+c)-regular interval-valued intuitionistic fuzzy
soft graph.
Therefore,
t d2 (a1) = u+c, β a1 β V
βΉ d2 (a1) + | D+
A (a1) - T+
A (a1) | = u+c, β a1 β V
βΉ d2 (a1) + | c1 βc2 | = u+c, β a1 β V
βΉ d2 (a1) + c = u+c, β a1 β V
βΉ d2 (a1) = u, β a1 β V
Hence,
G = (G*, FΝ, KΝ, (A, B)) is a (2, u)-regular interval-valued intuitionistic fuzzy soft graph.
Hence (i) and (ii) are equivalent. Conversely assume that (i) and (ii) are equivalent i.e., suppose
(G*, FΝ, KΝ, (A, B)) is (2, u)-regular interval-valued intuitionistic fuzzy soft graph and also a
totally (2, u+c))-regular interval-valued intuitionistic fuzzy soft graph.
Where, c = | c1 βc2 |.
Thus t d2 (a1) = u+c and d2 (a1) = u, β a1 β V
βΉ d2 (a1) + | D+
A (a1) - T+
A (a1) | = u+c and d2 (a1) = u, β a1 β V
βΉ | D+
A (a1) - T+
A (a1) | = c = | c1 βc2 |, β a1 β V
βΉ D+
A (a1) = c1 and T+
A (a1) = c2, β a1 β V
3. (B).Regularity on isomorphic interval-valued intuitionistic fuzzy soft graph
Definition: 3. (b) (1)
Let G1 = (G*, FΝ, KΝ, (A1, B1)) and G2 = (G*, FΝ, KΝ, (A2, B2)) be two interval-valued
intuitionistic fuzzy soft graph on (V1, E1) and (V2, E2) respectively.
A bijective function f: A1βA2 is called interval-valued intuitionistic fuzzy soft
morphism or f-morphism of interval-valued intuitionistic fuzzy soft graph if there exists some
positive real number u1 and u2 such that
(i)DA2 (f (a1)) = u1 DA1 a1 and TA2 (f(a1)) = u1 TA1 a1 , β a1 β V1
(ii)DB2 (f (a1), f (a2)) = u2 DB1 (a1, a2) and TB2 (f (a1), f (a2)) = u2 TB1 (a1, a2), β a1 β V1.
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In these cases f is called (u1, u2) f- interval-valued intuitionistic morphism on G1 over G2 when u1
= u2 = u then we say it is u-f- interval-valued intuitionistic morphism on G1 over G2.
Definition: 3. (b) (2)
A co-weak isomorphism from G1 to G2 is a map h: A1βA2 which is bijective
homomorphism that satisfies DB1 (a1, a2) = DB2 (h (a1), h (a2)) and
TB1 (a1, a2) = TB2 (h (a1), h (a2)), β a1, a2 β A.
A weak isomorphism from G1 to G2 is map h: A1βA2 which is bijective homomorphism that
satisfies DA1 (a1) = DA2 (h (a1)) and DA1 (a1) = TA2 (h (a1)), β a1 a2 β A.
Theorem: 3. (b) (1)
Let S be the set of all interval-valued intuitionistic fuzzy soft graphs. Now, define
the relation G1 β G2 when G1 is (u1, u2) f- interval-valued intuitionistic fuzzy soft morphism on
G2 where u1, u2 are any non-zero real numbers and G1, G2 β S.
Now for any identity morphism G1 over G1 is an one-one mapping and hence β² β β² is reflexive.
Let G1 β G2, then there exists a (u1, u2) -interval-valued intuitionistic fuzzy soft
morphism from G1 to G2 for some non-zero u1 and u2.
DA2 (f (a1)) = u1 DA1 a1 and TA2 (f (a1)) = u1 TA1 a1 , β a1 β V1
DB2 (f (a1), f (a2)) = u2 DB1 (a1, a2) and TB2 (f (a1), f (a2)) = u2 TB1 (a1, a2), β a1 ,a2 β V1
Consider f-1
: G1βG2. Let b1, b2 β V2.
As f-1
is bijective, b1= f (a1), b2= f (a2), for some a1 a2 β V1
Now
DA1(f-1
(b1)) = DA1(f-1
(f(a1) =DA1(a1) = 1
π’1 DA2f(a1) =
1
π’1 DA2(b1)
TA1(f-1
(b1)) = TA1(f-1
(f(a1) =TA1(a1) = 1
π’1 TA2f(a1) =
1
π’1 TA2(b1)
DB1(f-1
(b1), f-1
(b2)) = DB1(f-1
(f(a1), f-1
(f(a2)) =DB1(a1,a2) = 1
π’2 DB2 (f(a1),f(a2))
= 1
π’2 DB2 (b1,b2)
TB1(f-1
(b1), f-1
(b2)) = TB1(f-1
(f(a1), f-1
(f(a2)) =TB1(a1,a2) = 1
π’2 TB2 (f(a1),f(a2))
= 1
π’2 TB2 (b1,b2)
Thus there exists( 1
π’1 ,
1
π’2 ) f- interval-valued intuitionistic fuzzy soft morphism from G2 to G1.
Therefore G2 β G1 and hence βββ is symmetric.
Let G1 β G2 and G2 β G3
Thus there exist two interval-valued intuitionistic fuzzy soft morphism say (u1, u2)-f and
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(u2, u3)-g such that f is interval-valued intuitionistic fuzzy soft morphism from G1 to G2 and g is
interval-valued intuitionistic fuzzy soft morphism from G2 to G3 for non-zero u1, u2, u3, u4.So,
DA3 g (b1) = u3 DA2 (b1) and TA3 g (b1) = u3 TA2 (b1) ,β b1 β V2 and
DB3(g(b1),g(b2)) = u4 DB2 (b1,b2) and TB3 (g(b1),g(g2)) = u4 TB2 (b1,b2) ,β (b1, b2) β E2.
Let h=gβf: G1β G3. Now,
DA3 (h (a1)) = DA3 ((gβf) (a1)) = DA3 (g (f (a1))) = u3 DA2 (f (a1)) = u3 u1 DA1 (a1)
TA3 (h (a1)) = TA3 ((gβf) (a1)) = TA3 (g (f (a1))) = u3 TA2 (f (a1)) = u3 u1 TA1 (a1)
DB3 (h (a1), h (a2)) = DB3 ((gβf) (a1), (gβf) (a2)) = DB3 (g (f (a1)), g (f (a2))) = u4 DB2 (f (a1), f (a2))
= u4 u2 DB1 (a1, a2)
TB3 (h (a1), h (a2)) = TB3 ((gβf) (a1), (gβf) (a2)) = TB3 (g (f (a1)), g (f (a2))) = u4 TB2 (f (a1), f (a2))
= u4 u2 TB1 (a1, a2)
Thus, there exists (u3 u1, u4 u2) h- interval-valued intuitionistic fuzzy soft morphism from G1
over G3. Therefore, G1 β G3 hence βββ is transitive.
So, the relation f- interval-valued intuitionistic fuzzy soft morphism is an equivalence relation in
the collection of all interval-valued intuitionistic fuzzy soft graph.
Theorem: 3. (b) (2)
Let G1 and G2 be two IVIFSGβS such that G1 is (u1, u2) interval-valued intuitionistic
fuzzy soft morphic to G2 for some non-zero u1 and u2. The image of strong edge in G1 is strong
edge in G2 if and only if u1 = u2.
Proof
Let (a1, a2) be strong edge in G1 such that f (a1) , f(a2)
is also strong edge in G2.
Now, as G1 β G2
u2 DB1 (a1, a2) = DB2 (f (a1), f (a2)) = DA2 f (a1) β§ DA2 f (a2) = u1 {DA1 (a1) β§ DA1 (a2)}
= u1 DB1(a1, a2), β a1 β V1.
Hence, u2 DB1 (a1, a2) = u1 DB1 (a1, a2), β a1 β V1 ........................................................................... (1)
Similarly, u2 TB1 (a1, a2) = TB2 (f (a1), f (a2)) = TA2 f (a1) β¨ TA2 f (a2) = u1 {TA1 (a1) β§ TA1 (a2)}
= u1 TB1 (a1,a2), β a1 β V1.
Hence, u2 TB1 (a1, a2) = u1 TB1 (a1, a2), β a1 β V1 ............................................................................. (2)
Equation (1) and (2) holds. i.e., u1 = u2. Hence the theorem.
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Theorem: 3. (b) (3)
If an IVIFSG G1 is co- weak isomorphic to IVIFSG G2 and if G1 is regular then G2 is
regular.
Proof
As IVIFSG G1 is co- weak isomorphic to IVIFSG G2, there exists a co weak
isomorphism h: G1β G2 which is bijective that satisfies
DA1 (a1) β€ DA2 (h (a1)) and TA1 (a1) β₯ TA2 (h (a1)).
It also satisfies,
DB1 (a1, a2) = DB2 (h (a1), h (a2)) and
TB1 (a1, a2) = TB2 (h (a1), h (a2)), β a1, a2 β V1.
As G1 is regular, for a1 β V.
βπ1β π2 π·B+ (π1, π2) = constant.
π2βπ£1
βπ1β π2 TB+ (π1, π2) = constant.
π2βπ£1
ββ(π1)β β(π2) π·B2 (h (a1), h (a2)) = βπ1β π2 π·B+ (π1, π2) = constant.
And ββ(π1)β β(π2) TB2 (h (a1), h (a2)) = βπ1β π2 TB
+ (π1, π2) = constant.
Therefore G2 is regular.
Theorem: 3. (b) (4)
Let G1 and G2 be two IVIFSG. If G1 is weak isomorphic to G2 and if G1 is strong
then G2 is strong.
Proof
As an IVIFSG G1 be weak isomorphic with an IVIFSG G2, there exists a weak
isomorphic h: G1βG2 which is bijective that satisfies
DA1 (a1) = DA2 (h (a1)) and TA1 (a1) = TA2 (h (a1)),
DB1 (a1, a2) β€ DB2 (h (a1), h (a2)) and TB1 (a1, a2) β₯ TB2 (h (a1), h (a2)) β a1, a2 β V1.
As G1 is strong, DB1 (a1, a2) = min DA1 (a1), DA1 (a2) and TB1 (a1, a2) = max TA1 (a1), TA1 (a2)
DB2 (h (a1), h (a2)) β€ DB1 (a1, a2) = min {DA1 (a1), DA1 (a2)} = min {DA2 h (a1), DA2 h (a2)}
By definition, DB2 (h (a1), h (a2)) β€ min {DA2 h (a1), DA2 h (a2)}
Therefore, DB2 (h (a1), h (a2)) = min {DA2 h (a1), DA2 h (a2)} Similarly,
TB2 (h (a1), h (a2)) β₯ TB1 (a1, a2) = max {TA1 (a1), TA1 (a2)} = max {TA2 h (a1), TA2 h (a2)}
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And by definition,TB2 (h (a1), h (a2)) β₯ max {TA2 h (a1), TA2 h (a2)}.
Therefore, TB2 (h (a1), h (a2)) = max {TA2 h (a1), TA2 h (a2)}
Thus G2 is strong.
4. Conclusion
A regular interval-valued intuitionistic fuzzy soft graph has numerous applications
in the modeling of real life system where the level of information inherited in the system varies
with respect to time and have a different level of precision and hesitation. Most of the actions in
real life are time dependent, symbolic models used in the expert system are more effective than
traditional one. In this paper, we introduced the concept of a regular interval-valued intuitionistic
fuzzy graph and obtained some properties over it. In future, we can extend this concept to bipolar
fuzzy soft graphs, hyper graphs and in some more areas of graph theory.
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