More on the Annihilator-Ideal Graph of a Commutative Ring

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.