A Generalization Of Co-Maximal Graph Of Commutative Rings

Let R be a commutative ring with 1. Let G(R) be a graph with vertices as elements of R, where two distinct vertices a and b are adjacent if and only if aR+bR = eR for some non-zero idempotent e in R. In this paper, we establish a relation between completeness of the graph G(R) and regularity of the ring R. For a finite commutative ring R with 1, we show that the chromatic number of G(R) is equal to the number of regular elements in R. Moreover, we characterize some graph theoretic properties of G(R) and finally we characterize Eulerian property of the graph G(R).