Exploring Geometrical Properties of Annihilator Intersection Graph of Commutative Rings

Let Λ denote a commutative ring with unity and D(Λ) denote a collection of all annihilating ideals from Λ. An annihilator intersection graph of Λ is represented by the notation AIG(Λ). This graph is not directed in nature, where the vertex set is represented by D(Λ)*. There is a connection in the form of an edge between two distinct vertices ς and ϱ in AIG(Λ) iff Ann(ςϱ)≠Ann(ς)∩Ann(ϱ). In this work, we begin by categorizing commutative rings Λ, which are finite in structure, so that AIG(Λ) forms a star graph/2-outerplanar graph, and we identify the inner vertex number of AIG(Λ). In addition, a classification of the finite rings where the genus of AIG(Λ) is 2, meaning AIG(Λ) is a double-toroidal graph, is also investigated. Further, we determine Λ, having a crosscap 1 of AIG(Λ), indicating that AIG(Λ) is a projective plane. Finally, we examine the domination number for the annihilator intersection graph and demonstrate that it is at maximum, two.