Some graph invariants of Co-Intersection graph of ideals of a commutative ring

This study presents a structural investigation of the Co-Intersection graph, denoted by [Formula: see text], which is defined on the set of all non-trivial ideals of a commutative ring [Formula: see text]. In this undirected graph, two distinct vertices [Formula: see text] and [Formula: see text] are adjacent whenever the sum of the corresponding ideals satisfies [Formula: see text]. Although co-intersection graphs were previously introduced and examined by Hoseini and Talebi [The co-intersection graphs of ideals of rings, Ann. Pure Appl. Math. 28(2) (2023) 63–71], the existing literature does not provide a classification of finite commutative rings according to the clique number of [Formula: see text]. This work fills this gap by completely characterizing all finite commutative rings [Formula: see text] for which the clique number of [Formula: see text]. Furthermore, explicit formulas are derived for the first and second Zagreb indices of [Formula: see text], where [Formula: see text] is a prime and [Formula: see text], thereby contributing new topological invariants to the study of co-intersection graphs associated with principal ideal rings. These results expand earlier studies and explain more clearly how properties of a ring are connected to the features of its co-intersection graph.