Some results on a supergraph of the comaximal ideal graph of a commutative ring

The rings considered in this article are commutative with identity which admit at least two maximal ideals. We denote the set of all maximal ideals of a ring R by Max(R) and we denote the Jacobson radical of R by J (R). Let R be a ring such that |Max(R)| ≥ 2. Let I(R) denote the set of all proper ideals of R. In this article, we associate an undirected graph denoted by INC(R) with a subcollection of ideals of R whose vertex set is {I ∈ I(R)|I /⊆ J (R)} and two distinct vertices I₁, I₂ are adjacent in INC(R) if and only if I₁ /⊆ I₂ and I₂ /⊆ I₁ (that is, I₁ and I₂ are not comparable under the inclusion relation). The aim of this article is to investigate the interplay between the graph-theoretic properties of INC(R)