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 Gamma(I)(R), is the graph whose vertices are the set {x is an element of R\I vertical bar xy is an element of I for some y is an element of R\I} with distinct vertices x and y adjacent if and only if xy is an element of I . In the case I = 0, Gamma(0) (R), denoted by Gamma(R), is the zero-divisor graph which has well known results in the literature. In this article we explore the relationship between Gamma(I)(R) congruent to Gamma(J) (S) and Gamma(R/I) congruent to Gamma(S/J). We also discuss when Gamma(I)(R) is bipartite. Finally we give some results on the subgraphs and the parameters of Gamma(I)(R).