On zero-divisor graphs of finite rings

The zero-divisor graph of a ring R is defined as the directed graph Gamma (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, Gamma (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 Gamma (M-n (R)) similar or equal to Gamma (M-m (S)), then n = m, vertical bar R vertical bar vertical bar S vertical bar, and Gamma (R) similar or equal to Gamma (S). (c) 2007 Elsevier Inc. All rights reserved.