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 QGammaI(R) of R with respect to I which is an undirected graph with vertex set V=ainRbackslashsqrtI:$$abinI$forsome$binRbackslashsqrtI and two distinct vertices a and b are adjacent if and only if abinI. 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 QGammaI(R).