  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.
