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

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