Generalized zero-divisor graph of $*$-rings

Abstract

Let $R$ be a ring with involution $*$ and $Z^*(R)$ denotes the set of all non-zero zero-divisors of $R$. We associate a simple (undirected) graph $\Gamma'(R)$ with vertex set $Z^*(R)$ and two distinct vertices $x$ and $y$ are adjacent in $\Gamma'(R)$ if and only if $x^ny^*=0$ or $y^nx^*=0$, for some positive integer $n$. We find the diameter and girth of $\Gamma'(R)$. The characterizations are obtained for $*$-rings having $\Gamma'(R)$ 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 $\Gamma'(R\times R)$ is disconnected if and only if $R$ is an integral domain.