Generalized zero-divisor graph of $\ast$-rings

Abstract

Let $R$ be a ring with involution $\ast$ and $Z^{\ast }(R)$ denotes the set of all non-zero zero-divisors of $R$. We associate a simple (undirected) graph ${\Gamma }^{\prime }(R)$ with vertex set $Z^{\ast }(R)$ and two distinct vertices $x$ and $y$ are adjacent in ${\Gamma }^{\prime }(R)$ if and only if $x^n{y}^{\ast }=0$ or $y^n{x}^{\ast }=0$, for some positive integer $n$. We find the diameter and girth of ${\Gamma }^{\prime }(R)$. The characterizations are obtained for $\ast$-rings having ${\Gamma }^{\prime }(R)$ a connected graph, a complete graph, and a star graph. Further, we have shown that for a ring $R$, there is an involution on $R\times R$ such that ${\Gamma }^{\prime }(R\times R)$ is disconnected if and only if $R$ is an integral domain.