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Electronic Notes in Discrete Mathematics 63 (2017) 517–524 www.elsevier.com/locate/endm
Signed Zero-Divisor Graph Deepa Sinha 1 Deepakshi Sharma 2 Bableen Kaur 3 Department of Mathematics South Asian University Akbar Bhawan, Chanakyapuri New Delhi-110021, India
Abstract Let R be a finite commutative ring with unity (1 = 0) and let Z(R)∗ be the set of non-zero zero-divisors of R. We associate a (simple) graph Γ(R) to R with vertices as elements of R and for distinct x, y ∈ R, the vertices x and y are adjacent if and only if xy = 0. Further, its signed zero-divisor graph is an ordered pair ΓΣ (R) := (Γ(R), σ), where for an edge ab, σ(ab) is ‘+’ if a ∈ Z(R)∗ or b ∈ Z(R)∗ and ‘−’ otherwise. This paper aims at gaining a deeper insight into signed zero-divisor graph by investigating properties like, balancing, clusterability, sign-compatibility and consistency. Keywords: finite commutative ring, zero-divisors, signed graph, negation signed graph, balancing, clusterability, sign-compatible, consistent.
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Introduction
The idea of a zero-divisor graph of a commutative ring R was introduced by I. Beck [1], where he was mainly interested in colorings of R. They let all 1 2 3
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elements of R be vertices and had distinct vertices x and y adjacent if and only if xy = 0. The zero-divisor graphs Γ(Z4 ) and Γ(Z6 ) are shown in Figure 1.
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(a) Γ(Z4 )
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Fig. 1. Zero-Divisor Graphs
Given a graph G, a signature σ on G is a mapping that assigns to each edge of G either a positive or a negative sign. The graph G equipped with a signature σ is called a signed graph, denoted by S := (G, σ), where G = (V, E) is an underlying graph and σ : E −→ {+, −} is the signature that labels each edge of G either by ‘+’ or ‘−’. The edge which receives the positive (respectively, negative) sign is called a positive (respectively, negative) edge. A signed graph is an all-positive (respectively, all-negative) if all its edges are positive (respectively, negative); further, it is said to be homogeneous if it is either an all-positive or an all-negative and hetrogeneous otherwise. The negation η(S) of a signed graph S is a signed graph obtained from S by negating the sign of every edge of S. One of the fundamental concepts in the theory of signed graph is that of balance. Harary [8] introduced the concept of balanced signed graphs for the analysis of social networks, in which a positive edge stands for a positive relation and a negative edge represents a negative relation. They have been rediscovered many times because they come up naturally in many unrelated areas. The following is the characterization of a balanced signed graph from Cartwright and Harary [3]. Lemma 1.1 [3](Structure Theorem) An signed graph S is balanced if and only if its vertex set can be partitioned into two subsets V1 and V2 , one of them may be empty, such that any edge joining two vertices within the same subset is positive, while any edge joining two vertices in different subsets is negative. The social groups corresponding to V1 and V2 in Lemma 1.1 may be re-
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garded as subgroups of similar ideologies in the social system. Thus, a social system is called balanced if all relations between people are positive. No additional restriction applies if neither a positive nor a negative relation (i.e., indifference) exists between individuals in the same or different subgroups. However, suppose it is not possible to partition the given signed graph according to Harary bipartition, then such a system is unbalanced, implying that it is unnatural and susceptible to inner tension. A related but alternate theory is to anticipate a clustering of people into factions or clusters (not necessarily two) where the only relations occurring between individuals in the same cluster are positive and the only relations occurring between individuals in different clusters are negative. Social system possessing this property are called clusterable. A signed graph is said to be clusterable if its vertex set can be partitioned into pairwise disjoint subsets, called clusters, such that every positive edge joins vertices within the subset and every negative edge joins vertices between subsets. We shall call such a partition a Davis partition, after its originator [5] or a clustering [4]. Theorem 1.2 [4] A signed graph S is clusterable if and only if no cycle in S contains exactly one negative edge. This paper insights into the study of zero-divisor graph of a commutative ring in the realm of signed graph using algebraic approach. Definition 1.3 A signed zero-divisor graph is an ordered pair ΓΣ (R) := (Γ(R), σ), where Γ(R) is the zero-divisor graph of a commutative ring R and for an edge ab of ΓΣ (R), σ is defined as ⎧ ⎨ + if a ∈ Z(R)∗ or b ∈ Z(R)∗ , σ(ab) = ⎩ − otherwise. The signed zero-divisor graphs of Z4 and Z6 are shown in Figure 2, in which the positive edges are drawn as solid line segment and the negative edges as broken line segment. A graph G is called a labeled graph if there exists a mapping μ from the vertex set of the graph G to the set {+, −}. A marked graph is an ordered pair Gμ := (G, μ), where G = (V, E) is an underlying graph an μ : V (G) −→ {+, −} is the labelling that labels each vertex of G either ‘+’ or ‘−’. A marked signed graph is an ordered pair Sμ = (S, μ), where S = (G, σ) is a signed graph and μ : V (G) −→ {+, −} is a marking of S. A signed graph S is sign-compatible if there exists a marking μ of its
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1 (a) ΓΣ (Z4 )
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Fig. 2. Signed Zero-Divisor graphs
vertices such that the end vertices of every negative edge receive ‘−’ sign in μ and no positive edge in S has both of its ends assigned ‘−’ sign in μ. A cycle Z in a marked signed graph Sμ is said to be consistent if it contains an even number of negative vertices. A given marked signed graph S is said be consistent if every cycle in it is consistent [2]. Unless mentioned otherwise, all rings considered in this paper are finite and commutative with unity (1 = 0). All rings have at least two elements. As usual, the ring of integers modulo n is denoted by Zn and finite field with p elements by Fp . For terminology and notation in graph theory and abstract algebra, we refer the reader to the standard textbooks [7] and [6], respectively.
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Some Properties on ΓΣ (R)
In this section, we show that ΓΣ (R) is always connected and has small diameter and girth. We denote the order of Γ(R) and ΓΣ (R) by |Γ(R)| and |ΓΣ (R)|, respectively. We start with some examples which motivates later results. Example 2.1 Below are the zero-divisor graphs for several rings with |ΓΣ (R)| ≤ 7. Upto isomorphism, each graph may be realized as ΓΣ (R) by precisely the following rings: (a) Z2 (b) Z3 (c) F4 (d) Z4 or Z2 [x]/x2 (e) Z2 × Z2 (f ) Z5 (g) Z6
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(h) Z7 Note that these examples show that nonisomorphic rings may have the same zero-divisor graph and that the zero-divisor graph does not detect nilpotent elements.
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Fig. 3. Examples showing all possible signed zero-divisor graphs with |ΓΣ (R)| ≤ 7
If the diam(G) and g(G) represents the diameter and girth of the graph G, respectively, then we have the following results. Theorem 2.2 Let R be a commutative ring with unity. Then (i) Γ(R) is finite if and only if R is finite.
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(ii) Γ(R) is connected and diam(Γ(R)) ≤ 2. (iii) If Γ(R) contains a cycle, then g(Γ(R)) = 3. Proof. (i) The proof of this part is straight forward. (ii) Let x, y ∈ R be distinct elements. If either of x = 0 or y = 0, then diam(x, y) = 1. Assume that both x and y are nonzero. Case 1 If x, y ∈ R−Z(R), then x−0−y is a path of length 2, thus d(x, y) = 2. Case 2 If x, y ∈ Z(R)∗ . If xy = 0, then d(x, y) = 1. If xy is nonzero, then x − 0 − y is a path of length 2, thus d(x, y) = 2 Case 3 If x ∈ R − Z(R) and y ∈ Z(R)∗ , then x − 0 − y is a path of length 2, thus d(x, y) = 2. Hence d(x, y) ≤ 2, and thus diam(Γ(R)) ≤ 2. (iii) Let us suppose Γ(R) contains a cycle. Then it will always contain a triangle, since in this case there exist at least two nonzero zero-divisors a, b ∈ R such that ab = 0. And hence 0−a−b−0 is the required triangle. Hence g(Γ(R)) = 3. 2
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Main Results
In this section, we have characterized the rings R whose signed zero-divisor graphs and its negation are balanced, clusterable, sign-compatible and consistent. Remark 3.1 For a finite commutative ring with unity R, its signed zerodivisor graph ΓΣ (R) is always balanced and clusterable. Theorem 3.2 For a finite commutative ring with unity R, its negation signed zero-divisor graph η(ΓΣ (R)) is balanced if and only if |Z(R)∗ | ≤ 1. Proof. Let R be a finite commutative ring with unity. Towards proving the necessity part, assume that η(ΓΣ (R)) is balanced. To show |Z(R)∗ | ≤ 1. Let us suppose |Z(R)∗ | > 1. Then there exist at least two nonzero zerodivisors, say a, b ∈ R such that ab = 0. Then 0 − a − b − 0 is an all-positive triangle in ΓΣ (R). Clearly, this triangle will be a negative triangle in η(ΓΣ (R)), which is a contradiction. Hence |Z(R)∗ | ≤ 1. For the sufficiency part, let |Z(R)∗ | ≤ 1.
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(i) If |Z(R)∗ | = 0, then ΓΣ (R) is an all-negative star graph trivially. Thus η(ΓΣ (R)) is an all-positive star graph. It follows that η(ΓΣ (R)) is trivially balanced. (ii) If |Z(R)∗ | = 1, then there exist exactly one nonzero zero-divisor, say a ∈ R. Then 0a is a positive edge in ΓΣ (R) and all other edges incident to 0 are negative (i.e., ΓΣ (R) is a star graph centered at “0” with exactly one positive edge). Then η(ΓΣ (R)) is another star graph centered at “0” with exactly one negative edge which makes η(ΓΣ (R)) balanced. 2 Remark 3.3 For a finite commutative ring with unity R, its negation signed zero-divisor graph η(ΓΣ (R)) is always clusterable. For the given signed zero-divisor graph, we can always mark the vertices such that the end vertices of every negative edge receive ‘−’ sign and no positive edge has both of its ends asigned ‘−’ by marking non-zero zero-divisors ‘+’ and other vertices by ‘−’ sign. And by marking the zero-divisors ‘−’ and all other vertices ‘+’, we can conclude that the negation of signed zero-divisor graph is also sign-compatible. Remark 3.4 For a finite commutative ring with unity R, its signed zerodivisor graph ΓΣ (R) and its negation graph η(ΓΣ (R)) are always sign-compatible. Further, by marking zero-divisors ‘+’ in signed zero-divisor graph, all cycles will contain even number of negative vertices and hence we can conclude that signed zero-divisor graph for any finite commutative ring with unity is consistent. And again with the same marking, its negation graph is also consistent. Remark 3.5 For a finite commutative ring with unity R, its signed zerodivisor graph ΓΣ (R) and its negation graph η(ΓΣ (R)) are always consistent.
References [1] Beck, I., Coloring of commutative rings, Journal of Algebra 116 (1988), 208– 226. [2] Beineke, L. W., and F. Harary, Consistent graphs with signed points, Rivista di mathematica per le scienze economiche e sociali 1 (1978), 81–88. [3] Cartwright, D., and F. Harary, Structural balance: a generalization of heider’s theory, Psychological review 63 (1956), 277.
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[4] Chartrand, G., Graphs and Mathematical Models, Prindle Weber and Schmidt (1977). [5] Davis, J. A., Clustering and structural balance in graphs, Human Relations 20 (1967), 181–187. [6] Dummit, D. S., and R. M. Foote, “Abstract Algebra,” John Wiley and Sons, 2004. [7] Harary, F., “Graph Theory,” Addison-Wesley, Reading, MA, 1969. [8] Harary, F., On the notion of balance of a signed graph, Michigan Mathematical Journal 2 (1953), 143–146.
