Let R be a commutative ring and let Γ(Zn) be the zero divisor graph of a commutative ring R, whose vertices are non-zero zero divisors of Zn, and such that the two vertices u, v are adjacent if n divides uv. In this paper, we introduce the concept of fuzzy zero zivisor graph in a commutative ring and also discuss the some special cases of Γf (Z2p), Γf (Z3p), Γf (Z5p), Γf (Z7p) and Γf (Zpq). Throughout this paper we denote the Fuzzy Zero Divisor Graph(FZDG) by Γf (Zn).Publisher's Versio

Kuppan, A.

Sankar, J. Ravi

TWMS Journal of Applied and Engineering Mathematics

English

Isik University Academic Open Access

TWMS J. App. and Eng. Math. V.11, Special Issue, 2021, pp. 42-50FUZZY ZERO DIVISOR GRAPH IN A COMMUTATIVE RINGA. KUPPAN1, J. RAVI SANKAR1, §Abstract. Let R be a commutative ring and let Γ(Zn) be the zero divisor graph of acommutative ring R, whose vertices are non-zero zero divisors of Zn, and such that thetwo vertices u, v are adjacent if n divides uv. In this paper, we introduce the concept offuzzy zero zivisor graph in a commutative ring and also discuss the some special casesof Γf (Z2p), Γf (Z3p), Γf (Z5p), Γf (Z7p) and Γf (Zpq). Throughout this paper we denotethe Fuzzy Zero Divisor Graph(FZDG) by Γf (Zn).Keywords: Fuzzy graph, Zero divisor graph, Fuzzy zero divisor graph(FZDG).AMS Subject Classification: 05C25, 05C69.1. IntroductionThe zero divisor graph is very useful to find the algebraic structures and propertiesof rings. The idea of a zero divisor graph of a commutative ring was introduced by I.Beck’s in [2]. Given a ring R, let G(R) denote the graph whose vertex set is R, such thatdistinct vertices r and s are adjacent provided that rs = 0. I.Beck’s main interest was thechromatic number χ(G(R)) of the graph G(R).Rosenfeld [7] defined a fuzzy graph as a graph that consists of vertices and edges withmembership value in the interval [0,1]. More specifically, he defined a fuzzy graph as apair G = (σ, µ) of functions σ : S → [0, 1] and µ : S × S → [0, 1] where for all x, y ∈ Swe have µ(x, y) = µ(y, x) and µ(x, y) = σ(x) ∧ σ(y) with ∧ denoting the minimum. Thegeneral terminology, notation everything based on the papers [ [1], [3] − [6]].In this paper, an attempt to combine the two concepts: Fuzzy graph theory and zerodivisor graph of a commutative ring together by introducing a new concept called fuzzyzero divisor graph of commutative ring. Finally, we discuss some specal cases of Γf (Z2p),Γf (Z3p), Γf (Z5p), Γf (Z7p) and Γf (Zpq).2. PreliminariesDefinition 2.1. [8] A fuzzy graph as a pair G = (σ, µ) of functions σ : S → [0, 1] andµ : S×S → [0, 1] where for all x, y ∈ S we have µ(x, y) = µ(y, x) and µ(x, y) = σ(x)∧σ(y)with ∧ denoting the minimum.1 Department of Mathematics, Vellore Institute of Technology, Vellore, Tamil Nadu, 632014, India.e-mail: kuppanmyname@gmail.com; ORCID: https://orcid.org/0000-0001-7874-0971.e-mail: ravisankar.j@vit.ac.in; ORCID: https://orcid.org/0000-0001-9094-503X.§ Manuscript received: October 19, 2019; accepted: April 8, 2020.TWMS Journal of Applied and Engineering Mathematics V.11, Special Issue © Işık University,Department of Mathematics, 2021; all rights reserved.42A. KUPPAN, J. RAVI SANKAR: FUZZY ZERO DIVISOR GRAPH IN A COMMUTATIVE RING 43Definition 2.2. [1] Let R be a commutative ring (with 1) and let Z(R) be its set ofzero-divisors. We associate a (simple) graph Γ(R) to R with vertices Z(R)∗ = Z(R)− 0,the set of nonzero zero-divisor of R, and for distinct x, y ∈ Z(R)∗ the vertices x and yare adjacent if and only if xy = 0. Thus Γ(R) is the empty graph if and only if R is anintegral domain.3. Fuzzy Zero Divisor GraphDefinition 3.1. An fuzzy zero divisor graph(FZDG) is of the form G =< V, Vf , Ef >then the vertex set of non-zero zero divisor graph isΓ(Zn) = V = {(u, v) : uv = 0 (multiplication modulo n),∀u, v ∈ V } (1)Let p and q are any prime numbers with p < q. ThenVf ={V f1 ∪ Vf2 | ∀ Vf1 , Vf2 ∈ (0, 1)}such that Vf : V → (0, 1) (2)where V f1 ={1p ,2p ,3p , ...,p−1p}and V f2 ={1q ,2q ,3q , ...,q−1q}, 0 < Vf < 1.Efpq ={Efq , Ef2q, Ef3q, ..., Efpq | ∀ Efjq ∈ (0, 1]}such that Ef : V → (0, 1] (3)where j = 1, 2, 3, ..., p, 0 < Ef ≤ 1.If any one of the condition is not satisfied, then the graph G is not an FZDG.Definition 3.2. A fuzzy graph G : (Vf , Ef ) is said to be a fuzzy star graph if Γ(Zn) is azero divisor graph where n=2p and p > 2.Theorem 3.1. [4] For (Z2p), where p is any prime number then γc(Γ(Z2p)) = 1. Also,if n=8,9 then γc(Γ(Zn)) = 1.Theorem 3.2. [4] In (Z3p) where p is any prime with p > 3, then γc(Γ(Z3p)) = 2.Theorem 3.3. [4] If p > 5 is any prime, then γc(Γ(Z5p)) = 2.Theorem 3.4. [4] For any graph (Z7p) where p is any prime with p > 7, then γc(Γ(Z7p)) =2.Theorem 3.5. [4] If p and q are distinct primes and q > p, then γc(Γ(Zpq)) = 2.Theorem 3.6. If n = 2p where p is any prime and p > 2 then Γf (Z2p) be the non-zerofuzzy zero divisor graph is Kf1,p−1 fuzzy star graph.Proof. Let Γ(Z2p) be a non-zero zero divisor graph. Then the vertex set of non-zero zerodivisor graph V = {(u, v) : uv = 0 (multiplication modulo n),∀u, v ∈ V }. Take two dis-tinct vertex sets V1 and V2 in Γ(Z2p) where V1 = {2} and V2 = {2, 4, 6, ..., 2(p− 1)} thenclearly we know that a vertex 2 ∈ V1 adjacent to all the vertices in V2.Clearly, Γ(Z2p) is isomorphic with K1,p−1.Case(i): If Vf is fuzzy vertex set. The vertex set of fuzzy zero divisor graph Vf is partitionin to two vertex subsets namely, Vf1 and Vf2 . LetVf1 ={1p,2p,3p, ...,p− 1p}44 TWMS J. APP. AND ENG. MATH. V.11, SPECIAL ISSUE, 2021where p is any prime number with p > 2.That is,Vf1 ={Np| N = 1, 2, 3, ..., p− 1}with Vf1 : V → (0, 1). Let Vf2 ={12}, with Vf2 : V → (0, 1)Since,Vf = Vf1 ∪ Vf2Vf ={1p,2p,3p, ...,p− 1p}∪{12}={1p,2p,3p, ...,p− 1p,12}= {Vf1 ∪ Vf2} : V → (0, 1)Vf =: V → (0, 1)Hence Vf be the fuzzy vertex set.Case(ii): If Ef is fuzzy edge set. Let as take any two vertices u, v ∈ V (Γ(Z2p)) andΓ(Z2p) = {(u, v) : uv = 0 multiplication modulo n,∀u, v ∈ V } .Let u = 2(p − 1) and v = p then uv = 2(p − 1).p = 2p(p − 1). Clearly, 2p must divides2p(p− 1), then there exist a edge connect between u and v.Let Ef be a collection of edges.Ef ={1,12,13, ...,1p− 1},where p is any prime number with p > 2.Ef ={1N| N = 1, 2, 3, ..., p− 1}.Thus, clearly Ef : V → (0, 1].Clearly we know that every vertex in Vf1 is adjacent to all the vertices in Vf2 . Hence thegraph Kf1,p−1 is fuzzy star graph.Figure 1. Γf (Z22)A. KUPPAN, J. RAVI SANKAR: FUZZY ZERO DIVISOR GRAPH IN A COMMUTATIVE RING 45Theorem 3.7. If n = 3p where p is any prime and p > 3 then Γf (Z3p) be the non-zerofuzzy zero divisor graph is Kf2,p−1 fuzzy complete bipartite graph.Proof. Let p is any prime number with greater than 3. Then the vertex set of Γ(Z3p) isV = {(u, v) : uv = 0 multiplication modulo n,∀u, v ∈ V }. Take two distinct vertex setsV1 and V2 in Γ(Z3p) where V1 = {p, 2p} and V2 = {3, 6, 9, ..., 3(p− 1)}. Every vertex in V1is adjacent to all the vertices in V2. Clearly, Γ(Z3p) is a complete bipartite graph, namelyK2,p−1.Let Vf be a fuzzy vertex set. Let us show that Vf : V → (0, 1).Fuzzy vertex set Vf is partition in to two vertex subsets, namely Vf1 and Vf2 .LetVf1 ={1p,2p,3p, ...,p− 1p},where p is any prime with p > 3.That is,Vf1 ={Np| N = 1, 2, 3, ..., p− 1},with Vf1 : V → (0, 1).Let Vf2 ={13 ,23}with Vf2 → (0, 1)Clearly,Vf = Vf1 ∪ Vf2={(1p,2p,3p, ...,p− 1p)∪(13,23)}={1p,2p,3p, ...,p− 1p,13,23}where p is any prime with p > 3.Thus, Vf = {Vf1 ∪ Vf2} : V → (0, 1) which implies that Vf : V → (0, 1)Hence, Vf is a fuzzy vertex set.Let Ef be a fuzzy edge set. Let as show that Ef : V → (0, 1].Let as take any two vertices u, v ∈ V (Γ(Z3p)). Edge set E(G) defined byE(G) = {(u, v) : uv = 0 (multiplication modulo n), ∀u, v ∈ V } . Let u = p, v = 2p or u =2p, v = p. Then uv = 2p× p = 2p2 which does not divide by 3p.Therefore u and v are non-adjacent vertices in Γ(Z3p). Let x be any other vertex in Γ(Z3p)such that ux = vx = 0. That is the remaining vertices in Γ(Z3p) are adjacent to both uand v.46 TWMS J. APP. AND ENG. MATH. V.11, SPECIAL ISSUE, 2021Let Efp , Ef2p ∈ Ef , where Efp and Ef2p are collection of fuzzy edges from p and 2p respec-tively. since vertex p and 2p are the adjacent to all the vertices in Γ(Z3p).Efp ={1,12,13, ...,1p− 1}Ef2p ={12,14,16, ...,12(p− 1))}Ef ={Efp , Ef2p}Ef =[{1,12,13, ...,1p− 1},{12,14,16, ...,12(p− 1))}]where p is any prime number with p > 3.which implies Ef : V → (0, 1). Hence Ef is a fuzzy edge set. Clearly, Γf (Z3p) = (V, Vf , Ef )be a fuzzy zero divisor graph in Γf (Z3p) and we know that every vertex in Vf1 is adjacentto all the vertices in Vf2 . Hence that the graph Kf2,p−1 is fuzzy complete bipartite graph.Figure 2. Γf (Z15)Theorem 3.8. If n = 5p where p is any prime and p > 5 then Γf (Z5p) be the non-zerofuzzy zero divisor graph is Kf4,p−1 fuzzy complete bipartite graph.Proof. Let p is any prime number, greater than 5. Then the vertex set of Γ(Z5p) isV = {(u, v) : uv = 0 multiplication modulo n,∀u, v ∈ V } where n=3p. Take two distinctvertex sets V1 and V2 in Γ(Z5p) where V1 = {p, 2p, 3p, 4p} and V2 = {5, 10, 15, ..., 5(p−1)}.Every vertex in V1 is adjacent to all the vertices in V2. Clearly, Γ(Z5p) complete bipartitegraph, namely K4,p−1.Let Vf be a fuzzy vertex set. Let as show that Vf : V → (0, 1). Fuzzy vertex set Vf ispartition in to two vertex subsets, namely Vf1 and Vf2 .LetVf1 ={1p,2p,3p, ...,p− 1p},where p is any prime number with p > 5.That is,Vf1 ={Np| N = 1, 2, 3, ..., p− 1},with Vf1 : V → (0, 1).A. KUPPAN, J. RAVI SANKAR: FUZZY ZERO DIVISOR GRAPH IN A COMMUTATIVE RING 47Let Vf2 ={15 ,25 ,35 ,45}with Vf2 → (0, 1)Clearly,Vf = Vf1 ∪ Vf2={(1p,2p,3p, ...,p− 1p)∪(15,25,35,45)}={1p,2p,3p, ...,p− 1p,15,25,35,45}where p is any prime number with p > 3.Thus, Vf = {Vf1 ∪ Vf2} : V → (0, 1) which implies that Vf : V → (0, 1)Hence, Vf is a fuzzy vertex set.Let Ef is fuzzy edge set. Let as show that Ef : V → (0, 1]. Let as take any two verticesu, v ∈ V (Γ(Z5p)). Let u = 2p and v = 3p in Γ(Z5p) then 5p does not divides uv = 6p2,which implies that no two vertices of V1 and V2 are adjacent.Let Efp , Ef2p, Ef3p, Ef4p ∈ Ef , where Efp , Ef2p, Ef3p and Ef4p are collection of fuzzy edgesfrom p, 2p, 3p and 4p respectively. Since vertex p, 2p, 3p and 4p are the adjacent to allthe vertices in V2(Γ(Z5p)).Efp ={1,12,13, ...,1p− 1}Ef2p ={12,14,16, ...,12(p− 1)}E3p ={13,16,19, ...,13(p− 1)}Ef4p ={14,18,112, ...,14(p− 1)}Ef = [Efp , Ef2p, Ef3p, Ef4p]Ef =[{1,12,13, ...,1p− 1},{12,14,16, ...,12(p− 1)},{13,16,19, ...,13(p− 1)},{14,18,112, ...,14(p− 1)}]where p is any prime with p > 5, which implies Ef : V → (0, 1]. Hence, Ef is a fuzzy edgeset. Clearly, Γf (Z5p) = (V, Vf , Ef ) be a fuzzy zero divisor graph in Γf (Z5p). Every vertexin Vf1 is adjacent to all the vertices in Vf2 . Hence that the graph Kf4,p−1 is fuzzy completebipartite graph.Theorem 3.9. If n = 7p where p is any prime and p > 7 then Γf (Z7p) be the non-zerofuzzy zero divisor graph is Kf6,p−1 fuzzy complete bipartite graphProof. From the theorem 3.6, theorem 3.7 and theorem 3.8, Γf (Z7p) is a fuzzy completebipartite graph namely Kf6,p−1.48 TWMS J. APP. AND ENG. MATH. V.11, SPECIAL ISSUE, 2021Figure 3. Γf (Z35)Theorem 3.10. If n = pq where p and q are distinct prime numbers and q > p thenΓf (Zpq) be the non-zero fuzzy zero divisor graph is Kfp−1,q−1 fuzzy complete bipartite graph.Proof. Let p and q are any prime numbers and p < q. Then the vertex set of Γ(Zpq)is V = {(u, v) : uv = 0 multiplication modulo n,∀u, v ∈ V } where n=pq. Taketwo distinct vertex sets V1 and V2 in Γ(Zpq) where V1 = {p, 2p, 3p, ..., p(q − 1)} andV2 = {q, 2q, 3q, ..., (p− 1)q}.Every vertex in V1 adjacent to all the vertices in V2. Clearly, Γ(Zpq) complete bipartitegraph, namely Kp−1,q−1.Let Vf is fuzzy vertex set. Let as show that Vf : V → (0, 1). Fuzzy vertex set Vf partitionin to two sets, namely Vf1 and Vf2 .LetVf1 ={1p,2p,3p, ...,(p− 1)p}where p and q are any prime numbers with p < q. That is,Vf1 ={Np| N = 1, 2, 3, ..., p− 1}with Vf1 : V → (0, 1). LetVf2 ={1q,2q,3q, ...,(q − 1)q}with Vf2 : V → (0, 1). where p and q are any prime numbers with p < q.Vf = Vf1 ∪ Vf2={1p,2p,3p, ...,(p− 1)p}∪{1q,2q,3q, ...,(q − 1)q}={1p,2p,3p, ...,(p− 1)p,1q,2q,3q, ...,(q − 1)q}where p and q are any prime numbers with p < q. Thus, Vf = {Vf1 ∪ Vf2} : V → (0, 1)which implies that Vf : V → (0, 1). Hence Vf is a fuzzy vertex set.A. KUPPAN, J. RAVI SANKAR: FUZZY ZERO DIVISOR GRAPH IN A COMMUTATIVE RING 49Let Ef is fuzzy edge set. Let as show that LetEfp ={1,12,13, ...,1q − 1}Ef2p ={12,14,16, ...,12(q − 1)}E3p ={13,16,19, ...,1p(q − 1)}...Efpq ={1p,12p,13p, ...,1p(q − 1)}Ef = [Efp , Ef2p, Ef3p, ..., Efpq]Ef =[{1,12,13, ...,1q − 1},{12,14,16, ...,12(q − 1)},{13,16,19, ...,1p(q − 1)},...,{1p,12p,13p, ...,1p(q − 1)}]where p and q are any prime numbers with p < q. which implies that Ef : V → (0, 1].Hence Ef is a fuzzy edge set.Clearly, Γf (Zpq) = (V, Vf , Ef ) is called fuzzy zero divisor graph. Therefore every vertex inVf1 is adjacent to all the vertices in Vf2 . Hence that the graph Kfp−1,q−1 is fuzzy completebipartite graph. 4. ConclusionIn this paper, we have defined the Fuzzy Zero Divisor Graph of a commutative ring.Also established some special cases of Γf (Z2p), Γf (Z3p), Γf (Z5p), Γf (Z7p) and Γf (Zpq).In future we will study some more properties and applications of FZDG.References[1] Anderson, D. F. and Livingston, P. S., (1999), The Zero-Divisor Graph of a Commutative Ring, J.Algebra., 217, pp. 434-447[2] Beck, I., (1998), Coloring of Commutative Ring, J. Algebra., 116, pp. 208-226[3] Ravi Sankar, J and Meena, S., (2012), Changing and unchanging the Domination Number of a com-mutative ring, International Journal of Algebra, 6, No-27, pp. 1343 - 1352.[4] Ravi Sankar, J and Meena, S., (2012), Connected Domination Number of a commutative ring, Inter-national Journal of Mathematical Research, 5, No-1, pp. 5 - 11.[5] Ravi Sankar, J and Meena, S., (2013), On Weak Domination in a Zero Divisor Graph, InternationalJournal of Applied Mathematics, 26, No-1, pp. 83 - 91.[6] Ravi Sankar, J and Meena, S., (2014), Notes on the Strong Domination in a Zero Divisor Graph,Advances in Algebra, 7, No-1, pp. 1-7.[7] Rosenfeld, A.: Fuzzy graphs. In: Zadeh, L.A., Fu, K.S., Shimura, M. (eds.) Fuzzy Sets and TheirApplications, pp. 77–95. Academic Press, New York (1975)[8] Zadeh, L. A., (1965), Fuzzy Sets, Information Science., 8, pp. 338-353.50 TWMS J. APP. AND ENG. MATH. V.11, SPECIAL ISSUE, 2021A.Kuppan was born in Tamil Nadu, India. He obtained his M.Sc.(Mathematics)degree in 2017 from Islamiah college, Vaniyambadi, Tamil Nadu andM.Phil.(Mathematics) degree from Department of Mathematics, Islamiah col-lege, Vaniyambadi in 2018. Currently, he is a research scholar (Ph.D) in theDepartment of Mathematics, Vellore Institute of Technology, Vellore, Tamil Nadu,India. His research interest focus mainly on Algebra, Graph theory and Fuzzy graph.J.Ravi Sankar is an Assistant Professor (Senior) of Mathematics in Vellore Instituteof Technology, Vellore, Tamil Nadu, India. He is Completed Ph.D. from ManonmaniumSundaranar University. He has published over 45 research papers in various interna-tional and national journals and currently guiding four students for Ph. D. an area ofinterest of research includes various topics in Algebra, Fuzzy Mathematics, DiscreteMathematics, Applied Mathematics, Graph Theory like Domination, Coloring.

Fuzzy zero divisor graph in a commutative ring

http://acikerisim.isikun.edu.tr/xmlui/bitstream/11729/3019/1/vol.11-Special_Issue-05.pdf

Fuzzy zero divisor graph in a commutative ring

Abstract

Similar works

Full text

Available Versions

Isik University Academic Open Access