The weakly zero-divisor graph of a commutative ring

Abstract

 Let $R$ be a commutative ring with identity, and let $Z(R)$ be the set of zero-divisors of $R$. The weakly zero-divisor graph of $R$ is the undirected (simple) graph $W\Gamma(R)$ with vertex set $Z(R)^*$, and two distinct vertices $x$ and $y$ are adjacent if and only if there exist $r\in \operatorname{ann}(x)$ and $s\in \operatorname{ann}(y)$ such that $rs=0$. It follows that $W\Gamma(R)$ contains the zero-divisor graph $\Gamma(R)$ as a subgraph. In this paper, the connectedness, diameter, and girth of $W\Gamma(R)$ are investigated. Moreover, we determine all rings whose weakly zero-divisor graphs are star. We also give conditions under which weakly zero-divisor and zero-divisor graphs are identical. Finally, the chromatic number of $W\Gamma(R)$ is studied.