retractions and homomorphisms on some operations of...

Research ArticleRetractions and Homomorphisms on SomeOperations of Graphs

M Abu-Saleem

Department of Mathematics Faculty of Science Al-Balqa Applied University Salt 19117 Jordan

Correspondence should be addressed to M Abu-Saleem mohammedabusaleem2005yahoocom

Received 18 May 2018 Revised 9 August 2018 Accepted 13 September 2018 Published 1 October 2018

Academic Editor Francisco Balibrea

Copyright copy 2018 M Abu-Saleem This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The aim of the present article is to introduce and study a new type of operations on graph namely edge graphThe relation betweenthe homomorphisms and retractions on edge graphs is deducedThe limit retractions on the edge graphs are presented Retractionson a finite number of edge graphs are obtained

1 Introduction and Preliminaries

Graph theory is rapidly moving into the mainstream ofmath-ematics The prospects of further development in algebraicgraph theory and important link with computational theoryindicate the possibility of the subject quickly emerging atthe forefront of mathematics Its scientific and engineeringapplications especially to communication science computertechnology and system theory have already been accordeda place of pride in applied mathematics Graphs serve asmathematical models to analyze successfully many concretereal-world problems A certain problem in physics chem-istry genetics psychology sociology and linguistics canbe formulated as problems in graph theory Also manybranches of mathematics such as game theory group theorymatrix theory probability and topology have interactionswith graph theory Some puzzles and various problems of apractical nature have been instrumental in the developmentof various topics in graph theoryThe theory of acyclic graphswas developed for solving problems of electrical networksand the study of trees was developed for enumeratingisomers of organic compounds This paper describes theoperation of a graph from the viewpoint of an identification[1ndash10]

A graph is an ordered119866 = (119881(119866) 119864(119866)) where119881(119866) = 120601119864(119866) is a set disjoint from 119881(119866) elements of 119881(119866) are calledthe vertices of119866 and elements of 119864(119866) are called the edges Agraph is connected if for every partition of its vertex set intotwo nonempty sets 119883 and 119884 there is an edge with one end in

119883 and one end in 119884 otherwise the graph is disconnected Agraph119867 is said to be a subgraph of a graph119866 if119881(119867) sube 119881(119866)and 119864(119867) sube 119864(119866) A graph in which each pair of distinctvertices is adjacent is called a complete graph A completegraph with n vertices is denoted by 119870119899 [11] The chromaticnumber 120594(119866) of a graph 119866 is the minimum number of colorsrequired for proper vertex coloring of 119866 A 119898minuscoloring of agraph119866 is a vertex coloring of119866 that uses atmost119898minuscolors Agraph 119866 is said to be119898-colorable if 119866 admits a proper vertexcoloring using at most 119898 colors [11] Let 119866 and 119867 be twographs A function 120601 119881(119866) 997888rarr 119881(119867) is a homomorphismfrom119866 to119867 if it preserves edges that is if for any edge [119906 V]of 119866 [120601(119906) 120601(V)] is an edge of 119867 [12] A retract of a graph 119866is a subgraph119867 of 119866 such that there exists a homomorphism119903 119866 997888rarr 119867 called retraction with 119903(119909) = 119909 for any vertex 119909of119867 [7] A core is a graph which does not retract to a propersubgraph [12]

2 The Main Results

Aiming at our study we will introduce the following

Definition 1 Let 1198661 and 1198662 be two connected graphs where1198901 is an edge of 1198661 1198902 is an edge of 1198662 and 1198661 cap1198662 = 120601 thenwe define the edge graph 1198661or1198662 by gluing together the twoedges 1198901 and 1198902

Theorem 2 Let 1198661 and 1198662 be two connected graphs Then120594(1198661or1198662) = max120594(1198661) 120594(1198662)

HindawiJournal of MathematicsVolume 2018 Article ID 7328065 4 pageshttpsdoiorg10115520187328065

2 Journal of Mathematics

Proof Let 120594(1198661or1198662) = 119898 At that point there exists an119898-coloring 120603 of 1198661or1198662 Since 120603 assigns different colors toevery two adjacent vertices of 1198661 and 1198662 1198661 and 1198662 are 119898-colorable and so 120594(1198661) le 120594(1198661or1198662) and 120594(1198662) le 120594(1198661or1198662)Also using symmetry 120594(1198661) ge 120594(1198662) Beginning with anideal coloring of 120594(1198661) we can incorporate an ideal coloringof 120594(1198662) by exchanging a pair of color names to make thecoloring agree at two vertices of common edge graphs Thisproduces a proper coloring of 1198661or1198662

Theorem 3 The graphs 1198661 and 1198662 are subgraphs of 1198661or1198662Also for any tree 1198661 and 1198662 the graph 1198661or1198662 is also a tree

Proof The proof of this theorem is clear

Theorem4 Suppose that1198661 1198662 119866119899 are connected graphsthen there is a sequence of nontrivial retractions 119903119896 or119899119894=1

119866119894 997888rarr or119899119894=1

119866119894 119896 = 1 2 119899 where or119899119894=1

119866119894 are gluedalong the same edge such that 119903119896(or

119899

119894=1119866119894) is a proper subgraph

of or119899119894=1

119866119894

Proof Let 1199031 or119899

119894=1119866119894 997888rarr or119899

119894=1119866119894 be a retraction from or119899

119894=1119866119894

into itself and 1199031(or119899

119894=1119866119894) = 1198661or1198662or or1199031(119866119904) or119866119899 for

119904 = 1 2 119899 Since 1199031(119866119904) is a proper subgraph of 119866119904 itfollows that 1199031(or

119899

119894=1119866119894) is a proper subgraph of or119899

119894=1119866119894 Also

if 1199032(or119899

119894=1119866119894) = 1198661or1198662or sdot sdot sdot or1199032(119866119904)or sdot sdot sdot or1199032(119866119896)or sdot sdot sdot or119866119899 for

119896 = 1 2 119899 119904 lt 119896 and 1199032(119866119904) 1199032(119866119896) are subgraphs of119866119904 119866119896 respectively then 1199032(or

119899

119894=1119866119894) is a proper subgraph of

or119899119894=1

119866119894 Moreover by continuing this process if 119903119899(or119899

119894=1119866119894) =

or119899119894=1

119903119899(119866119894) then 119903119896(or119899

119894=1119866119894) is a proper subgraph ofor

119899

119894=1119866119894

Theorem 5 Let 1198661 and 1198662 be two graphs then there is ahomomorphism 119891 1198661 997888rarr 1198662 iff 1198662 is a retract of 1198661or1198662

Proof Let 119891 1198661 997888rarr 1198662 be a homomorphism Since 1198662is subgraph of 1198661or1198662 then there exists a homomorphism119903 1198661or1198662 997888rarr 1198662with 119903(119909) = 119909 for any vertex 119909 of 1198662 andso 1198662 is a retract of 1198661or1198662 Conversely assume that 1198662 is aretract of 1198661or1198662 thus 119903 1198661or1198662 997888rarr 1198662 is a homomorphismwith 119903(119909) = 119909 for any vertex 119909 of 1198662 and so there is ahomomorphism 119891 1198661 997888rarr 1198662

Theorem 6 Let 1198661 and 1198662 be connected graphs then 119870119899 is aretract of graph 1198661 or 1198662 iff 119870119899 retract of 1198661or1198662

Proof Suppose 1198661 and 1198662 are connected graphs and 119870119899 is aretract of graph 1198661 or 1198662 Then there is a homomorphism1199031 1198661 997888rarr 119870119899 such that 1199031(119909) = 119909 or a homomorphism1199032 1198662 997888rarr 119870119899 such that 1199032(119909) = 119909 for any vertex 119909 of 119870119899Since119870119899 is a core and it is subgraph of1198661 or1198662 it follows that119870119899 is subgraph of 1198661or1198662 and so there is a homomorphism119903 1198661or1198662 997888rarr 119870119899 such that 119903(119909) = 119909 for any vertex 119909 of119870119899 Conversely suppose 119870119899 is a retract 1198661or1198662 then 119870119899 issubgraph of1198661or1198662 and119870119899or119870119899 and so119870119899 is a retract of graphs1198661 or 1198662

Theorem 7 Let 119879 be any tree of size 119899 then there is asequence of nontrivial retractions 119903119894 119894 = 1 2 119899 such thatlim119899997888rarrinfin119903119899(119903119899minus1) (1199031(119879) = 1198702

+

H

H H

H

2CHC

2CHC

X

XX

X

C2CH

C2CH

(R)

(R) (R)

(R)

Figure 1

Proof Consider the following sequence of retractions1199031 119879 997888rarr 1198791 is nontrivial retraction where 1198791 is

subgraph of 119879 and 1 le 119878119894119911119890(1198791) le 119899 minus 11199032 1199031(119879) 997888rarr 1199031(1198791) where 1199031(1198791) is subgraph of 1199031(119879)

and 1 le 119878119894119911119890(1199031(1198791)) le 119899 minus 2119903119899 119903119899minus1(119903119899minus2) (1199031(119879) 997888rarr 119903119899minus1(119903119899minus2) (1199031(1198791)

where 119903119899minus1(119903119899minus2) (1199031(1198791) is subgraph of 119903119899minus1(119903119899minus2) (1199031(119879)and lim119899997888rarrinfin119903119899(119903119899minus1) (1199031(119879) is a tree of size 1 Thereforelim119899997888rarrinfin119903119899(119903119899minus1) (1199031(119879) = 1198962

Theorem 8 Suppose that 1198661 and 1198662 are connected graphsthen lim119899997888rarrinfin119903119899(1198661or1198662)) = lim119899997888rarrinfin119903119899(1198661)orlim119899997888rarrinfin119903119899(1198661)

Proof If 1198661 and 1198662 are connected graphsthen we get the following induced subgraphslim119899997888rarrinfin119903119899(1198661or1198662)) lim119899997888rarrinfin119903119899(1198661) lim119899997888rarrinfin119903119899(1198661)and each of them is isomorphic to 1198962 Since1198962 asymp 1198962or1198962 it follows that lim119899997888rarrinfin119903119899(1198661or1198662)) =lim119899997888rarrinfin119903119899(1198661)orlim119899997888rarrinfin119903119899(1198661)

3 Some Applications in Chemistryand Biology

(i) A polymer is composed of many repeating units calledmonomers Starch cellulose and proteins are natural poly-mers Nylon and polyethylene are synthetic polymers Poly-merization is the process of joiningmonomers Polymersmaybe formed by addition polymerization and one basic step inaddition polymerization is combination as in Figure 1 whichoccurswhen the polymerrsquos growth is stopped by free electronsfrom two growing chains that join and form a single chainThe following diagram depicts combination with the symbol(R) representing the rest of the chain This is a representationtype of connected two graphs into an edge graph

(ii) Peptide bonds constitute the representation of an edgegraph by linking two amino acids as in Figure 2 which is a

Journal of Mathematics 3

R

H

H H

H

C

C C

CCO

COOH COOH

COOH

NH R

R Rand

Peptide Bond

represent two typical amino acids

where NH2

NH2

NH2

Figure 2

H

N

C 0

H

H

C

0

H2R

0

H

H

N

C

H

H

C

0

1R

H

H

0

H

H

0waterwater

N

0

H

C0

H

H

N

C

H

H

C

0

1R

C

H 2R

(N terminus)amino terminus

(C terminus)carboxy terminus

Figure 3

representation graph of connected two typical amino acidsinto an edge graph

In Figure 3 peptide pond and formation hydrolysisFormation (top to bottom) and hydrolysis from bottomto top of a peptide bonds require conceptually loss andaddition respectively of a molecule of water The actualchemical synthesis and hydrolysis of peptide bonds in the cellare enzymatically controlled processes that in the synthesisnearly always occur on the ribosome and are directed by an

mRNA template The end of a polypeptide with the free ofamino group is known as the amino terminus (N terminus)and with the free carboxyl group is the carboxyl terminus (Cterminus) This is a representation of connected two graphsinto an edge graph

Data Availability

No data were used to support this study


Conflicts of Interest

The author declares that they have no conflicts of interest

References

[1] M Abu-Saleem ldquoFolding on the wedge sum of graphs and theirfundamental grouprdquo APPS Applied Sciences vol 12 pp 14ndash192010

[2] M Abu-Saleem ldquoDynamical manifold and their fundamentalgrouprdquoAdvanced Studies in ContemporaryMathematics vol 20no 1 pp 125ndash131 2010

[3] M Abu-Saleem ldquoConditional fractional folding of a manifoldand their fundamental grouprdquo Advanced Studies in Contempo-rary Mathematics vol 20 no 2 pp 271ndash277 2010

[4] M Abu-Saleem ldquoOn dynamical chaotic de Sitter spaces andtheir deformation retractsrdquo Proceedings of the Jangjeon Math-ematical Society Memoirs of the Jangjeon Mathematical Societyvol 14 no 2 pp 231ndash238 2011

[5] M Abu-Saleem ldquoOn the dynamical hyperbolic 3-spaces andtheir deformation retractsrdquo Proceedings of the Jangjeon Math-ematical Society Memoirs of the Jangjeon Mathematical Societyvol 15 no 2 pp 189ndash193 2012

[6] M Abu-Saleem ldquoOn chaotic homotopy grouprdquo AdvancedStudies in Contemporary Mathematics vol 23 no 1 pp 69ndash752013

[7] P Hell and J Nesetril Graphs and Homomorphisms vol 28Oxford University Press Oxford UK 2004

[8] S Pirzada and A Dharwadker ldquoApplications of GraphTheoryrdquoJKSIAM vol 11 no 4 pp 19ndash38 2007

[9] A T White Graphs Groups and Surfaces North-HollandPublishing Co Amsterdam Holland 1973

[10] R JWilson and J JWatkinsGraphs An IntroductoryApprouchA First Course in Discrete Mahematics Canada John Wiley andsons Inc 1990

[11] G Chartrand and P Zhang Chromatic Graph Theory Taylorand Francis group 2009

[12] G Hahn and C Tardif ldquoGraph homomorphisms structure andsymmetryrdquo in Graph symmetry vol 497 of NATO ASI Series C497 pp 107ndash166 Kluwer 1997

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Proof Let 120594(1198661or1198662) = 119898 At that point there exists an119898-coloring 120603 of 1198661or1198662 Since 120603 assigns different colors toevery two adjacent vertices of 1198661 and 1198662 1198661 and 1198662 are 119898-colorable and so 120594(1198661) le 120594(1198661or1198662) and 120594(1198662) le 120594(1198661or1198662)Also using symmetry 120594(1198661) ge 120594(1198662) Beginning with anideal coloring of 120594(1198661) we can incorporate an ideal coloringof 120594(1198662) by exchanging a pair of color names to make thecoloring agree at two vertices of common edge graphs Thisproduces a proper coloring of 1198661or1198662

Theorem 3 The graphs 1198661 and 1198662 are subgraphs of 1198661or1198662Also for any tree 1198661 and 1198662 the graph 1198661or1198662 is also a tree

Proof The proof of this theorem is clear

Theorem4 Suppose that1198661 1198662 119866119899 are connected graphsthen there is a sequence of nontrivial retractions 119903119896 or119899119894=1

119866119894 997888rarr or119899119894=1

119866119894 119896 = 1 2 119899 where or119899119894=1

119866119894 are gluedalong the same edge such that 119903119896(or

119899

119894=1119866119894) is a proper subgraph

of or119899119894=1

119866119894

Proof Let 1199031 or119899

119894=1119866119894 997888rarr or119899

119894=1119866119894 be a retraction from or119899

119894=1119866119894

into itself and 1199031(or119899

119894=1119866119894) = 1198661or1198662or or1199031(119866119904) or119866119899 for

119904 = 1 2 119899 Since 1199031(119866119904) is a proper subgraph of 119866119904 itfollows that 1199031(or

119899

119894=1119866119894) is a proper subgraph of or119899

119894=1119866119894 Also

if 1199032(or119899

119894=1119866119894) = 1198661or1198662or sdot sdot sdot or1199032(119866119904)or sdot sdot sdot or1199032(119866119896)or sdot sdot sdot or119866119899 for

119896 = 1 2 119899 119904 lt 119896 and 1199032(119866119904) 1199032(119866119896) are subgraphs of119866119904 119866119896 respectively then 1199032(or

119899

119894=1119866119894) is a proper subgraph of

or119899119894=1

119866119894 Moreover by continuing this process if 119903119899(or119899

119894=1119866119894) =

or119899119894=1

119903119899(119866119894) then 119903119896(or119899

119894=1119866119894) is a proper subgraph ofor

119899

119894=1119866119894

Theorem 5 Let 1198661 and 1198662 be two graphs then there is ahomomorphism 119891 1198661 997888rarr 1198662 iff 1198662 is a retract of 1198661or1198662

Proof Let 119891 1198661 997888rarr 1198662 be a homomorphism Since 1198662is subgraph of 1198661or1198662 then there exists a homomorphism119903 1198661or1198662 997888rarr 1198662with 119903(119909) = 119909 for any vertex 119909 of 1198662 andso 1198662 is a retract of 1198661or1198662 Conversely assume that 1198662 is aretract of 1198661or1198662 thus 119903 1198661or1198662 997888rarr 1198662 is a homomorphismwith 119903(119909) = 119909 for any vertex 119909 of 1198662 and so there is ahomomorphism 119891 1198661 997888rarr 1198662

Theorem 6 Let 1198661 and 1198662 be connected graphs then 119870119899 is aretract of graph 1198661 or 1198662 iff 119870119899 retract of 1198661or1198662

Proof Suppose 1198661 and 1198662 are connected graphs and 119870119899 is aretract of graph 1198661 or 1198662 Then there is a homomorphism1199031 1198661 997888rarr 119870119899 such that 1199031(119909) = 119909 or a homomorphism1199032 1198662 997888rarr 119870119899 such that 1199032(119909) = 119909 for any vertex 119909 of 119870119899Since119870119899 is a core and it is subgraph of1198661 or1198662 it follows that119870119899 is subgraph of 1198661or1198662 and so there is a homomorphism119903 1198661or1198662 997888rarr 119870119899 such that 119903(119909) = 119909 for any vertex 119909 of119870119899 Conversely suppose 119870119899 is a retract 1198661or1198662 then 119870119899 issubgraph of1198661or1198662 and119870119899or119870119899 and so119870119899 is a retract of graphs1198661 or 1198662

Theorem 7 Let 119879 be any tree of size 119899 then there is asequence of nontrivial retractions 119903119894 119894 = 1 2 119899 such thatlim119899997888rarrinfin119903119899(119903119899minus1) (1199031(119879) = 1198702

+

H

H H

H

2CHC

2CHC

X

XX

X

C2CH

C2CH

(R)

(R) (R)

(R)

Figure 1

Proof Consider the following sequence of retractions1199031 119879 997888rarr 1198791 is nontrivial retraction where 1198791 is

subgraph of 119879 and 1 le 119878119894119911119890(1198791) le 119899 minus 11199032 1199031(119879) 997888rarr 1199031(1198791) where 1199031(1198791) is subgraph of 1199031(119879)

and 1 le 119878119894119911119890(1199031(1198791)) le 119899 minus 2119903119899 119903119899minus1(119903119899minus2) (1199031(119879) 997888rarr 119903119899minus1(119903119899minus2) (1199031(1198791)

where 119903119899minus1(119903119899minus2) (1199031(1198791) is subgraph of 119903119899minus1(119903119899minus2) (1199031(119879)and lim119899997888rarrinfin119903119899(119903119899minus1) (1199031(119879) is a tree of size 1 Thereforelim119899997888rarrinfin119903119899(119903119899minus1) (1199031(119879) = 1198962

Theorem 8 Suppose that 1198661 and 1198662 are connected graphsthen lim119899997888rarrinfin119903119899(1198661or1198662)) = lim119899997888rarrinfin119903119899(1198661)orlim119899997888rarrinfin119903119899(1198661)

Proof If 1198661 and 1198662 are connected graphsthen we get the following induced subgraphslim119899997888rarrinfin119903119899(1198661or1198662)) lim119899997888rarrinfin119903119899(1198661) lim119899997888rarrinfin119903119899(1198661)and each of them is isomorphic to 1198962 Since1198962 asymp 1198962or1198962 it follows that lim119899997888rarrinfin119903119899(1198661or1198662)) =lim119899997888rarrinfin119903119899(1198661)orlim119899997888rarrinfin119903119899(1198661)

3 Some Applications in Chemistryand Biology

(i) A polymer is composed of many repeating units calledmonomers Starch cellulose and proteins are natural poly-mers Nylon and polyethylene are synthetic polymers Poly-merization is the process of joiningmonomers Polymersmaybe formed by addition polymerization and one basic step inaddition polymerization is combination as in Figure 1 whichoccurswhen the polymerrsquos growth is stopped by free electronsfrom two growing chains that join and form a single chainThe following diagram depicts combination with the symbol(R) representing the rest of the chain This is a representationtype of connected two graphs into an edge graph

(ii) Peptide bonds constitute the representation of an edgegraph by linking two amino acids as in Figure 2 which is a


R

H

H H

H

C

C C

CCO

COOH COOH

COOH

NH R

R Rand

Peptide Bond


where NH2

NH2

NH2

Figure 2

H

N

C 0

H

H

C

0

H2R

0

H

H

N

C

H

H

C

0

1R

H

H

0

H

H

0waterwater

N

0

H

C0

H

H

N

C

H

H

C

0

1R

C

H 2R



Figure 3




Data Availability





References




















Journal of












Journal of







Volume 2018






Volume 2018









R

H

H H

H

C

C C

CCO

COOH COOH

COOH

NH R

R Rand

Peptide Bond


where NH2

NH2

NH2

Figure 2

H

N

C 0

H

H

C

0

H2R

0

H

H

N

C

H

H

C

0

1R

H

H

0

H

H

0waterwater

N

0

H

C0

H

H

N

C

H

H

C

0

1R

C

H 2R



Figure 3




Data Availability





References




















Journal of












Journal of







Volume 2018






Volume 2018











References




















Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








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