Forbidden subgraphs in enhanced power graphs of finite groups

The enhanced power graph of a finite group is the graph whose vertex set is the set of the elements of this group, and two distinct vertices are adjacent if they generate a cyclic subgroup. In this paper, we classify all finite groups whose enhanced power graphs are split graphs and threshold graphs. We also classify all finite nilpotent groups whose enhanced power graphs are chordal graphs and Cographs. Finally, we give some families of non-nilpotent groups whose enhanced power graphs are chordal graphs and Cographs. The results partly answer a question was posed by Peter J. Cameron.