Co-maximal Subgroup Graphs of Finite Groups

For a finite group [Formula: see text], the co-maximal subgroup graph [Formula: see text] of [Formula: see text] is a graph whose vertices are proper subgroups of [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. The deleted co-maximal subgroup graph [Formula: see text] is obtained by removing isolated vertices from [Formula: see text]. Firstly, we provide necessary and sufficient conditions for [Formula: see text] to be connected; in particular, from the viewpoint of normal subgroups in [Formula: see text], we give some sufficient conditions for [Formula: see text] to be connected. Secondly, for a finite abelian group [Formula: see text] we prove that the diameter of [Formula: see text], [Formula: see text], is at most 3. Also, we characterize [Formula: see text] with [Formula: see text] for [Formula: see text] and we give characterizations for [Formula: see text] with [Formula: see text] being complete bipartite graphs and null graphs separately. Finally, we show that for the semidirect product [Formula: see text] of two finite cyclic groups, [Formula: see text] is connected and [Formula: see text].