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 Γ I (R), is the graph whose vertices are the set {x ∊ R | xy ∊ I for some y ∊ R} 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 Γ I (R) ≅ Γ J (S) and Γ(R/I) ≅ Γ(S/J). We also discuss when Γ I (R) is bipartite. Finally we give some results on the subgraphs and the parameters of Γ I (R).