Forbidden subgraphs in enhanced power graphs of finite groups

TL;DR

This paper classifies finite groups based on the properties of their enhanced power graphs, such as being split, threshold, chordal, or cographs, providing insights into their structural characteristics.

Contribution

It offers a comprehensive classification of finite groups with enhanced power graphs exhibiting specific graph-theoretic properties, addressing a question by Cameron.

Findings

Finite groups with split and threshold enhanced power graphs are classified.

Finite nilpotent groups with chordal and cograph enhanced power graphs are characterized.

Some non-nilpotent groups have enhanced power graphs that are chordal or cographs.

Abstract

The enhanced power graph of a group is the simple graph whose vertex set is consisted of all elements of the group, and whose any pair of vertices are adjacent if they generate a cyclic subgroup. In this paper, we classify all finite groups whose enhanced power graphs are split and threshold. 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. These results partly answer a question posed by Peter J. Cameron.