NILPOTENT GRAPHS OF MATRIX ALGEBRAS

Abstract

Let $R$ be a ring with unity. The undirected nilpotent graph of $R$, denoted by $Gamma_N(R)$, is a graph with vertex set ~$Z_N(R)^* = {0neq x in R | xy in N(R) for some y in R^*}$, and two <br />distinct vertices $x$ and $y$ are adjacent if and only if $xy in N(R)$, or equivalently, $yx in N(R)$, where $N(R)$ denoted the nilpotent elements of $R$. Recently, it has been proved that if $R$ is a left Artinian ring, then $diam(Gamma_{N}(R))leqslant 3$. In this paper, using the concept of rank over commutative rings, we investigate basic properties of undirected nilpotent graph of matrix algebras. Moreover, some result on undirected nilpotent graph of matrix algebras over commutative rings are given. For <br />instance, we prove that $Gamma_{N}(M_{n}(R))$ is not planar for all $ngeqslant 2$. Furthermore, we show that $diam(Gamma_{N}(R))leqslant diam(Gamma_{N}(M_{n}(R)))$ for an <br />Artinian commutative ring $R$. Also, we prove that $Gamma_{N}(M_{n}(R))congGamma_{N}(M_{n}(T(R)))$, where $T(R)$ be the total quotient ring of a commutative ring $R$