On the vertex connectivity of weakly zero-divisor graph of commutative rings

Abstract

The weakly zero-divisor graph $W\Gamma(R)$ of a commutative ring $R$ is the simple undirected graph whose vertices are nonzero zero-divisors of $R$, and two distinct vertices $x$, $y$ are adjacent if and only if there exists $w\in {\rm ann}(x)$ and $ z\in {\rm ann}(y)$ such that $wz =0$. In this paper, first we prove that the vertex connectivity of $W\Gamma(R)$ is equal to its minimum degree, where $R$ is either an Artinian ring or reduced ring. For any finite ring $R$, we obtain the vertex connectivity of $W\Gamma(R)$. Moreover, this paper characterizes all the vertices that attain the minimum degree of $W\Gamma(R)$.