Line graph characterization of the order supergraph of a finite group

TL;DR

This paper classifies finite groups based on whether their order supergraphs are line graphs or complements of line graphs, providing a structural understanding of these graph representations in group theory.

Contribution

It offers a complete classification of finite groups with order supergraphs that are line graphs or their complements, linking group properties to graph-theoretic structures.

Findings

Identified all finite groups with order supergraphs as line graphs.

Characterized groups whose order supergraphs are complements of line graphs.

Established connections between group order structures and graph line properties.

Abstract

The power graph $\mathcal{P}(G)$ is the simple undirected graph with group elements as a vertex set and two elements are adjacent if one of them is a power of the other. The order supergraph $\mathcal{S}(G)$ of the power graph $\mathcal{P}(G)$ is the simple undirected graph with vertex set $G$ in which two vertices $x$ and $y$ are adjacent if $o(x)\vert o(y)$ or $o(y)\vert o(x)$. In this paper, we classify all the finite groups $G$ such that the order supergraph $\mathcal{S}(G)$ is the line graph of some graph. Moreover, we characterize finite groups whose order supergraphs are the complement of line graphs.