Further study on forbidden subgraphs of power graph

TL;DR

This paper investigates forbidden subgraph classes in power graphs of groups, extending previous work by characterizing groups whose power graphs exclude specific complex subgraphs.

Contribution

It identifies groups with power graphs avoiding four new forbidden subgraph classes, including chain graphs and diamond-free graphs, and characterizes groups for certain other classes.

Findings

Characterized groups with chain graph power graphs.

Identified finite groups with diamond-free power graphs.

Determined groups with power graphs avoiding P5 and its complement.

Abstract

The undirected power graph (or simply power graph) of a group $G$, denoted by $P(G)$, is a graph whose vertices are the elements of the group $G$, in which two vertices $u$ and $v$ are adjacent if and only if either $u=v^m$ or $v=u^n$ for some positive integers $m$, $n$. Forbidden subgraph has a significant role in graph theory. In our previous work \cite{cmm}, we consider five important classes of forbidden subgraphs of power graph which include perfect graphs, cographs, chordal graphs, split graphs and threshold graphs. In this communication, we go even further in that way. This study, inspired by the articles \cite{celmmp,dong,ck}, examines additional $4$ significant forbidden classes, including chain graphs, diamond-free graphs, $\{P_{5}, \overline{P_{5}}\}$-free graphs and $\{P_{2}\cup P_{3}, \overline{P_{2}\cup P_{3}}\}$-free graph. The finite groups whose power graphs are chain graphs, diamond-free graphs, and $\{P_{2}\cup P_{3}, \overline{P_{2}\cup P_{3}}\}$-free graphs have been successfully identified in this work. In case of $\{P_{5}, \overline{P_{5}}\}$-free graphs, we completely determine all the nilpotent groups, direct product of two groups, finite simple groups whose power graph is $\{P_{5}, \overline{P_{5}}\}$-free.