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Let R be a commutative ring and Γ (R) be its zero-divisor graph. In this paper it is shown that for any finite commutative ring R, the edge chromatic number of Γ (R) is equal to the maximum degree of Γ (R), unless Γ (R) is a complete graph of odd order. In [D.F. Anderson, A. Frazier, A. Lauve, P.S. Livingston, in: Lecture Notes in Pure and Appl. Math., Vol. 220, Marcel Dekker, New York, 2001, pp. 61-72] it has been proved that if R and S are finite reduced rings which are not fields, then Γ (R) ≃ Γ (S) if and only if R ≃ S. Here we generalize this result and prove that if R is a finite reduced ring which is not isomorphic to ℤ2 × ℤ 2 or to ℤ6 and S is a ring such that Γ (R) ≃ Γ (S), then R ≃... 

The zero-divisor graph of a ring R is defined as the directed graph Γ (R) that its vertices are all non-zero zero-divisors of R in which for any two distinct vertices x and y, x → y is an edge if and only if x y = 0. Recently, it has been shown that for any finite ring R, Γ (R) has an even number of edges. Here we give a simple proof for this result. In this paper we investigate some properties of zero-divisor graphs of matrix rings and group rings. Among other results, we prove that for any two finite commutative rings R, S with identity and n, m ≥ 2, if Γ (Mn (R)) ≃ Γ (Mm (S)), then n = m, | R | = | S |, and Γ (R) ≃ Γ (S). © 2007 Elsevier Inc. All rights reserved  

Let Γ(R) be the zero-divisor graph of a commutative ring R. An interesting question was proposed by Anderson, Frazier, Lauve, and Livingston: For which finite commutative rings R is Γ (R) planar? We give an answer to this question. More precisely, we prove that if R is a local ring with at least 33 elements, and Γ(R) ≠ 0, then Γ(R) is not planar. We use the set of the associated primes to find the minimal length of a cycle in Γ(R). Also, we determine the rings whose zero-divisor graphs are complete r-partite graphs and show that for any ring R and prime number p, p ≥ 3, if Γ(R) is a finite complete p-partite graph, then Z(R) = p2, R = p3, and R is isomorphic to exactly one of the rings ℤp3,... 

In this paper, we deal with zero-divisor graphs of posets. We prove that the diameter of such a graph is either 1, 2 or 3 while its girth is either 3, 4 or ∞. We also characterize zero-divisor graphs of posets in terms of their diameter and girth  

In this paper, we characterize complete partite zero-divisor graphs of posets via the ideals of the posets. In particular, for complete bipartite zero-divisor graphs, we give a characterization based on the prime ideals of the posets  

In this paper, we discuss some recent results on graphs attached to rings. In particular, we deal with comaximal graphs, unit graphs, and total graphs. We then define the notion of cototal graph and, using this graph, we characterize the rings which are additively generated by their zero divisors. Finally, we glance at graphs attached to other algebraic structures