Ideal-based quasi zero divisor graph

Abstract

Let $R$ be a commutative ring with identity and $I$ a proper ideal of $R$. In this paper we introduce the ideal-based quasi zero divisor graph $Q\Gamma_{I}(R)$ of $R$ with respect to $I$ which is an undirected graph with vertex set $V=\{a\in R\backslash\sqrt{I}:$ $ab\in I$ for some $b\in R\backslash\sqrt{I}\}$ and two distinct vertices $a$ and $b$ are adjacent if and only if $ab\in I$. We study the basic properties of this graph such as diameter, girth, dominaton number, etc. We also investigate the interplay between the ring theoretic properties of a Noetherian multiplication ring $R$ and the graph-theoretic properties of $Q\Gamma_{I}(R)$.