Generalized zero-divisor graph of $*$-rings

TL;DR

This paper introduces a generalized zero-divisor graph for rings with involution, characterizes its properties, and explores conditions for connectivity and special graph structures.

Contribution

It defines a new class of zero-divisor graphs for $*$-rings and characterizes their properties, including diameter, girth, and conditions for specific graph types.

Findings

Determined the diameter and girth of the graph.

Characterized when the graph is connected, complete, or a star.

Established conditions for disconnected graphs in product 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.