Between the enhanced power graph and the commuting graph

TL;DR

This paper introduces the deep commuting graph of a finite group, which lies between the enhanced power graph and the commuting graph, and explores its properties and automorphisms.

Contribution

It defines the deep commuting graph, establishes conditions for it to coincide with existing graphs, and analyzes its automorphism group.

Findings

Deep commuting graph is contained in the commuting graph and contains the enhanced power graph.

Conditions are provided for when the deep commuting graph equals either the enhanced power graph or the commuting graph.

Automorphism group of the underlying group acts as automorphisms of the deep commuting graph.

Abstract

The purpose of this note is to define a graph whose vertex set is a finite group $G$, whose edge set is contained in that of the commuting graph of $G$ and contains the enhanced power graph of $G$. We call this graph the deep commuting graph of $G$. Two elements of $G$ are joined in the deep commuting graph if and only if their inverse images in every central extension of $G$ commute. We give conditions for the graph to be equal to either of the enhanced power graph and the commuting graph, and show that the automorphism group of $G$ acts as automorphisms of the deep commuting graph.