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Some Properties of the Nil-Graphs of Ideals of Commutative Rings

Let $R$ be a commutative ring with identity and ${\rm Nil}(R)$ be the set of nilpotent elements of $R$. The nil-graph of ideals of $R$ is defined as the graph $\mathbb{AG}_N(R)$ whose vertex set is $\{I:\ (0)\neq I\lhd R$ and there exists a non-trivial ideal $J$ such that $IJ\subseteq {\rm Nil}(R)\}$ and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ\subseteq {\rm Nil}(R)$. Here, we study conditions under which $\mathbb{AG}_N(R)$ is complete or bipartite. Also, the independence number of $\mathbb{AG}_N(R)$ is determined, where $R$ is a reduced ring. Finally, we classify Artinian rings whose nil-graphs of ideals have genus at most one.