The Complement of Enhanced Power Graph of a Finite Group

The enhanced power graph [Formula: see text] of a finite group [Formula: see text] is the simple undirected graph whose vertex set is [Formula: see text] and two distinct vertices [Formula: see text], [Formula: see text] are adjacent if [Formula: see text], [Formula: see text] for some [Formula: see text]. We give an affirmative answer to the question posed by Cameron which asks whether the complement of the enhanced power graph [Formula: see text] of a non-cyclic group[Formula: see text], [Formula: see text], has only one connected component apart from isolated vertices. We classify all finite groups [Formula: see text] for which [Formula: see text] is bipartite. We show that the graph [Formula: see text] is weakly perfect. Further, we study the subgraph [Formula: see text] of [Formula: see text] induced by all the non-isolated vertices of[Formula: see text]. We classify all finite groups[Formula: see text] with the graph [Formula: see text] being unicyclic and pentacyclic, and we prove the non-existence of finite groups [Formula: see text] such that the graph [Formula: see text] is bicyclic, tricyclic or tetracyclic. Finally, we characterize all finite groups [Formula: see text] for which the graph [Formula: see text] is outerplanar, planar, projective-planar and toroidal separately.