Central Sets in the Annihilating-Ideal Graph of Commutative Rings

Let \( R \) be a commutative ring with identity and \( \mathbb{A}^*(R) \) be the set of non-zero ideals with non-zero annihilators. The annihilating-ideal graph of \( R \) is defined as the graph \( \mathbb{AG}(R) \) with the vertex set \( \mathbb{A}^*(R) \) and two distinct vertices \( I_1 \) and \( I_2 \) are adjacent if and only if \( I_1 I_2 = (0) \). In this paper, we study some connections between the graph-theoretic properties of \( \mathbb{AG}(R) \) and algebraic properties of the commutative ring \( R \).