On the Difference Graph of power graphs of finite groups

TL;DR

This paper studies the difference graph between the enhanced power graph and the power graph of finite groups, classifying its properties for various group classes and specific groups like symmetric and alternating groups.

Contribution

It characterizes the difference graph's properties for finite groups, especially nilpotent groups, and classifies these graphs for symmetric and alternating groups.

Findings

Difference graphs are chordal, star, and threshold for certain groups.

Nilpotent groups have difference graphs with specific graph class properties.

Classified when difference graphs of symmetric and alternating groups are cograph, bipartite, etc.

Abstract

The power graph of a finite group $G$ is a simple undirected graph with vertex set $G$ and two vertices are adjacent if one is a power of the other. The enhanced power graph of a finite group $G$ is a simple undirected graph whose vertex set is the group $G$ and two vertices $a$ and $b$ are adjacent if there exists $c \in G$ such that both $a$ and $b$ are powers of $c$. In this paper, we investigate the difference graph $\mathcal{D}(G)$ of a finite group $G$, which is the difference of the enhanced power graph and the power graph of $G$ with all isolated vertices removed. We study the difference graphs of finite groups with forbidden subgraphs among other results. We first characterize an arbitrary finite group $G$ such that $\mathcal{D}(G)$ is a chordal graph, star graph, dominatable, threshold graph, and split graph. From this, we conclude that the latter four graph classes are equivalent for $\mathcal{D}(G)$. By applying these results, we classify the nilpotent groups $G$ such that $\mathcal{D}(G)$ belong to the aforementioned five graph classes. This shows that all these graph classes are equivalent for $\mathcal{D}(G)$ when $G$ is nilpotent. Then, we characterize the nilpotent groups whose difference graphs are cograph, bipartite, Eulerian, planar, and outerplanar. Finally, we consider the difference graph of non-nilpotent groups and determine the values of $n$ such that the difference graphs of the symmetric group $S_n$ and alternating group $A_n$ are cograph, chordal, split, and threshold.