On the nilpotent graph of a ring

Let R be a ring with unity. The nilpotent graph of R, denoted by Gamma_N(R), is a graph with vertex set Z_N(R)^* = {0 neq x in R mid xy in N(R) for some 0 neq y in R}; and two distinct vertices x and y are adjacent if and only if xy in N(R), where N(R) is the set of all nilpotent elements of R. Recently, it has been proved that if R is a left Artinian ring, then diam(Gamma_N(R)) leq 3. In this paper, we present a new proof for the above result, where R is a finite ring. We study the diameter and the girth of matrix algebras. We prove that if F is a field and n geq 3, then diam(Gamma_N(M_n(F))) = 2. Also, we determine diam (Gamma_N (M_2(F))) and classify all finite rings whose nilpotent graphs have diameter at most 3. Finally, we determine the girth of the nilpotent graph of matrix algebras.