  Let $R$ be a commutative ring with $1\neq 0$ and $\Bbb{A}(R)$ be the set of
ideals with nonzero annihilators. The annihilating-ideal graph of $R$ is
defined as the graph $\Bbb{AG}(R)$ with the vertex set $\Bbb{A}(R)^{*} =
\Bbb{A}(R)\setminus \{(0)\}$ and two distinct vertices $I$ and $J$ are adjacent
if and only if $IJ = (0)$. In this paper, we first study the interplay between
the diameter of annihilating-ideal graphs and zero-divisor graphs. Also, we
characterize rings $R$ when ${\rm gr}(\Bbb{AG}(R))\geq 4$, and so we
characterize rings whose annihilating-ideal graphs are bipartite. Finally, in
the last section we discuss on a relation between the Smarandache vertices and
diameter of $\Bbb {AG}(R)$.
