A note on a graph related to the comaximal ideal graph of a commutative ring

The rings considered in this article are commutative with identity which admit at least two maximal idealsý. ýThis article is inspired by the work done on the comaximal ideal graph of a commutative ringý. ýLet R be a ringý. ýWe associate an undirected graph to R denoted by \mathcal{G}(R)ý, ýwhose vertex set is the set of all proper ideals I of R such that I\not\subseteq J(R)ý, ýwhere J(R) is the Jacobson radical of R and distinct vertices I1ý, ýI2are adjacent in \mathcal{G}(R) if and only if I1∩ I2 = I1I2ý. ýThe aim of this article is to study the interplay between the graph-theoretic properties of \mathcal{G}(R) and the ring-theoretic properties of R.