Zero-Divisor Graph with Respect to an Ideal*

Let R be a commutative ring with nonzero identity and let I be an ideal of R. The zero-divisor graph of R with respect to I, denoted by ΓI(R), is the graph whose vertices are the set {x ∈ R \ I| xy ∈ I for some y ∈ R \ I} with distinct vertices x and y adjacent if and only if xy ∈ I. In the case I = 0, Γ0(R), denoted by Γ(R), is the zero-divisor graph which has well known results in the literature. In this article we explore the relationship between ΓI(R) ∼= ΓJ(S) and Γ(R/I) ∼= Γ(S/J). We also discuss when ΓI(R) is bipartite. Finally we give some results on the subgraphs and the parameters of ΓI(R).