Complement of prime ideals intersection graph of a ring

The complement of prime ideals intersection graph, denoted by [Formula: see text], of a commutative ring [Formula: see text] with unity [Formula: see text], is a graph whose vertex set is the set of all non-unit and non-zero elements of [Formula: see text]. Any two distinct vertices of [Formula: see text] are adjacent if and only if they both do not belong to the same prime ideal of [Formula: see text]. We establish the necessary and sufficient conditions on [Formula: see text] for which the graph [Formula: see text] is finite and then complete, respectively. We further examine the characteristics of [Formula: see text] when [Formula: see text] is acyclic, a tree, and a complete bipartite graph. We also examine its diameter, girth, clique number, maximum degree, and minimum degree. If [Formula: see text] constitutes a path, then the configuration of [Formula: see text] becomes significant. The study also examines the cut vertices, universal vertices, regularity, Hamiltonian properties, and the planarity of [Formula: see text].