Common zero-divisor graph over a commutative ring

Abstract

Let [Formula: see text] be a finite commutative ring with non-zero identity and [Formula: see text] be the set of all non-zero zero-divisors of [Formula: see text]. In this paper, for a non-empty subset [Formula: see text] of [Formula: see text], we introduce a new graph, denoted by [Formula: see text], with vertex set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if there is [Formula: see text] such that [Formula: see text]. Then, we investigate some properties of the graph [Formula: see text].