  Let $R$ be a commutative ring with identity and $\Bbb A (R)$ be the set of
ideals of $R$ with non-zero annihilator. The annihilator-ideal graph of $R$,
denoted by $A_{I} (R) $, is a simple graph with the vertex set $\Bbb
A(R)^{\ast} := \Bbb A (R) \setminus\lbrace (0) \rbrace $, and two distinct
vertices $I$ and $J$ are adjacent if and only if $\mathrm{Ann} _{R} (IJ) \neq
\mathrm{Ann} _{R} (I) \cup \mathrm{Ann} _{R} (J)$.
  In this paper, we study the affinity between the annihilator-ideal graph and
the annihilating-ideal graph $\Bbb A \Bbb G (R)$ (a well-known graph with the
same vertices and two distinct vertices $I,J$ are adjacent if and only if
$IJ=0$) associated with $R$. All rings whose $A_{I}(R) \neq \Bbb A \Bbb G (R)$
and $\mathrm{gr} (A_{I}(R)) =4$ are characterized. Among other results, we
obtain necessary and sufficient conditions under which $A_{I} (R)$ is a star
graph.
