A generalization of co-maximal graph of commutative rings
Let [Formula: see text] be a commutative ring with [Formula: see text]. Let [Formula: see text] be a graph with vertices as elements of [Formula: see text], where two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] for some non-zero idempotent [Formula: see text] in [Formula: see text]. In this paper, we establish a relation between completeness of the graph [Formula: see text] and regularity of the ring [Formula: see text]. For a finite commutative ring [Formula: see text] with [Formula: see text], we show that the chromatic number of [Formula: see text] is equal to the number of regular elements in [Formula: see text]. Moreover, we characterize some graph theoretic properties of [Formula: see text] and finally we characterize Eulerian property of the graph [Formula: see text].