On quasi-zero divisor graphs of non-commutative rings

Abstract

Let $R$ be an associative ring with identity. A ring $R$ is called reversible if $ab=0$, then $ba=0$ for $a,bin R$. The quasi-zero-divisor graph of $R$, denoted by $Gamma^*(R)$ is an undirected graph with all nonzero zero-divisors of $R$ as vertex set and two distinct vertices $x$ and $y$ are adjacent if and only if there exists $0neq rin R setminus (mathrm{ann}(x) cup mathrm{ann}(y))$ such that $xry=0$ or $yrx=0$. In this paper, we determine the diameter and girth of $Gamma^*(R)$. We show that the zero-divisor graph of $R$ denoted by $Gamma(R)$, is an induced subgraph of $Gamma^*(R)$. Also, we investigate when $Gamma^*(R)$ is identical to $Gamma(R)$. Moreover, for a reversible ring $R$, we study the diameter and girth of $Gamma^*(R[x])$ and we investigate when $Gamma^*(R[x])$ is identical to $Gamma(R[x])$.