Finite groups whose commuting graphs are line graphs

TL;DR

This paper classifies finite groups based on whether their commuting graphs or related subgraphs are line graphs or complements of line graphs, providing a comprehensive understanding of their structural properties.

Contribution

It offers a complete classification of finite groups whose commuting graphs or their variants are line graphs or their complements, extending previous graph-theoretic characterizations.

Findings

Identified all finite groups with commuting graphs as line graphs.

Classified groups with commuting graphs as complements of line graphs.

Extended classification to subgraphs obtained by removing certain vertices.

Abstract

The commuting graph ${\Gamma(G)}$ of a group $G$ is the simple undirected graph with group elements as a vertex set and two elements $x$ and $y$ are adjacent if and only if $xy=yx$ in $G$. By eliminating the identity element of $G$ and all the dominant vertices of $\Gamma(G)$, the resulting subgraphs of $\Gamma(G)$ are $\Gamma^*(G)$ and $\Gamma^{**}(G)$, respectively. In this paper, we classify all the finite groups $G$ such that the graph $\Delta(G) \in \{\Gamma(G), \Gamma^*(G), \Gamma^{**}(G)\}$ is the line graph of some graph. We also classify all the finite groups $G$ whose graph $\Delta(G) \in \{\Gamma(G), \Gamma^*(G), \Gamma^{**}(G)\}$ is the complement of line graph.