Quasi-zero-divisor graphs of non-commutative rings

Let $R$ be an associative ring with identity. 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 $0\neq r\in R \setminus \{ann(x) \cup ann(y)\}$ such that $xry=0$ or $yrx=0$. In this paper we study the propertis of $\Gamma^*(R)$. We show that the zero-divisor graph $\Gamma(R)$, is an induced subgraph of $\Gamma^*(R)$. Among other things, we show that % we completely determine connectedness, the diameter and the girth of $\Gamma^*(R)$. for a reversible ring $R$, the diameter of $\Gamma^*(R)$ is at most 2 and its girth is at most 4 provided that $\Gamma^*(R)$ has a cycle. We also study the properties of $\Gamma^*(R)$, when $R$ is reduced. Among other things investigate when $\Gamma^*(R)$ is a complete graph, star graph or when is identical to $\Gamma(R)$ .