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 $w\in \mathrm{ann}(x)$ and $z\in \mathrm{ann}(y)$ such that $wz=0$. In this paper, we obtain the vertex connectivity of $W\Gamma (R)$, where $R$ is an Artinian ring or reduced ring. Indeed, for these rings, we prove that the vertex connectivity of $W\Gamma (R)$ is equal to its minimum degree. This paper also characterizes all the vertices of minimum degree of $W\Gamma (R)$.