Zero-divisor graphs of non-commutative rings

In a manner analogous to the commutative case, the zero-divisor graph of a non-commutative ring R can be 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. We investigate the interplay between the ring-theoretic properties of R and the graph-theoretic properties of Γ(R). In this paper it is shown that, with finitely many exceptions, if R is a ring and S is a finite semisimple ring which is not a field and Γ(R) Γ(S), then R S. For any finite field F and each integer n 2, we prove that if R is a ring and Γ(R) Γ(Mn(F)), then R Mn(F). Redmond defined the simple undirected graph obtaining by deleting all directions on the edges in Γ(R). We classify all ring R whose is a complete graph, a bipartite graph or a tree.