On zero-divisor graphs of finite rings

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 xy=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).