Non co-maximal graph of subgroups of an abelian group

Let [Formula: see text] be a group. The co-maximal graph of [Formula: see text], denoted by [Formula: see text], is a graph whose vertices are nontrivial elements of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent in [Formula: see text] if and only if [Formula: see text]. In this paper, we study some connections between the graph-theoretic properties of this graph and some algebraic properties of groups. For instance, we characterized all groups whose co-maximal graphs are connected and we show that if [Formula: see text] has no isolated vertex, then [Formula: see text] is connected with diameter at most 2. Moreover, we classify all groups whose co-maximal graphs are complete. Also, some results on the clique number and chromatic number of a co-maximal graph are given. In addition, the co-maximal graphs of abelian groups are studied.