On chordality of the power graph of finite groups

TL;DR

This paper investigates which finite groups have chordal power graphs, characterizing direct products, classifying simple groups of Lie type, and showing sporadic groups are non-chordal, with most small groups having chordal power graphs.

Contribution

It provides a complete classification of finite groups with chordal power graphs, including simple groups of Lie type and direct products, and shows sporadic groups are non-chordal.

Findings

Direct product groups with chordal power graphs characterized.

All finite simple groups of Lie type with chordal power graphs identified.

Sporadic simple groups have non-chordal power graphs.

Abstract

A graph is called chordal if it forbids induced cycles of length 4 or more. In this paper, we attempt to identify the non-nilpotent groups whose power graph is a chordal graph (this question was raised by Cameron in [4]). In this direction, we characterise the direct product of finite groups having chordal power graphs. We classify all finite simple groups of Lie type whose power graph is chordal. Further, we prove that the power graph of a sporadic simple group is always non-chordal. In addition, we show that almost all groups of order up to 47 have chordal power graphs.