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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 $\{x \in R\setminus I|\ xy \in I\ {\rm forsome}\ y \in R\setminus I\}$ with distinct vertices $x$ and $y$
adjacent if and only if $xy \in I$. 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 ${\Gamma_I}(R)\cong
{\Gamma_J}(S)$ and ${\Gamma}(R/I)\cong {\Gamma}(S/J)$. We also discuss when ${\Gamma_I}(R)$ is bipartite. Finally we give some
results on the subgraphs and the parameters of ${\Gamma_I}(R)$.
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