The complement of enhanced power graph of a finite group

TL;DR

This paper investigates the properties of the complement of the enhanced power graph of finite groups, classifying groups based on graph properties like bipartiteness, cyclicity, and planarity, and answering a question posed by Cameron.

Contribution

It provides a complete classification of finite groups according to the structure and properties of the complement of their enhanced power graphs, including connectivity, bipartiteness, and planarity.

Findings

The complement of the enhanced power graph of a non-cyclic group has only isolated vertices and one connected component.

Classified all groups where the complement graph is bipartite, unicyclic, or pentacyclic.

Proved the non-existence of groups with bicyclic, tricyclic, or tetracyclic complement graphs.

Abstract

The enhanced power graph $\mathcal{P}_E(G)$ of a finite group $G$ is the simple undirected graph whose vertex set is $G$ and two distinct vertices $x, y$ are adjacent if $x, y \in \langle z \rangle$ for some $z \in G$. In this article, we give an affirmative answer of the question posed by Cameron [6] which states that: Is it true that the complement of the enhanced power graph $\bar{\mathcal{P}_E(G)}$ of a non-cyclic group $G$ has only one connected component apart from isolated vertices? We classify all finite groups $G$ such that the graph $\bar{\mathcal{P}_E(G)}$ is bipartite. We show that the graph $\bar{\mathcal{P}_E(G)}$ is weakly perfect. Further, we study the subgraph $\bar{\mathcal{P}_E(G^*)}$ of $\bar{\mathcal{P}_E(G)}$ induced by all the non-isolated vertices of $\bar{\mathcal{P}_E(G)}$. We classify all finite groups $G$ such that the graph is $\bar{\mathcal{P}_E(G^*)}$ is unicyclic and pentacyclic. We prove the non-existence of finite groups $G$ such that the graph $\bar{\mathcal{P}_E(G^*)}$ is bicyclic, tricyclic or tetracyclic. Finally, we characterize all finite groups $G$ such that the graph $\bar{\mathcal{P}_E(G^*)}$ is outerplanar, planar, projective-planar and toroidal, respectively.