A Note on a graph associated to a commutative ring

The rings considered in this article are commutative with identity. This article is motivated by the work on comaximal graphs of rings. In this article, with any ring R , we associate an undirected graph denoted by G(R) , whose vertex set is the set of all elements of R and distinct vertices x,y are joined by an edge in G(R) if and only if Rx∩Ry=Rxy . In Section 2 of this article, we classify rings R such that G(R) is complete and we also consider the problem of determining rings R such that χ(G(R))=ω(G(R))