# NILPOTENT GRAPHS OF MATRIX ALGEBRAS

Document Type : Research Paper

Author

Payame Noor University

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 ~$Z_N(R)^* = \{0\neq x \in R |\ xy \in N(R) \ for\ some\ y \in R^*\}$, and two
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
instance, we prove that $\Gamma_{N}(M_{n}(R))$ is not planar for all $n\geqslant 2$. Furthermore, we show that $\diam(\Gamma_{N}(R))\leqslant \diam(\Gamma_{N}(M_{n}(R)))$ for an
Artinian commutative ring $R$. Also, we prove that $\Gamma_{N}(M_{n}(R))\cong\Gamma_{N}(M_{n}(T(R)))$, where $T(R)$ be the total quotient ring of a commutative ring $R$

Keywords

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