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

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