Co-maximal subgroup graph characterized by forbidden subgraphs

TL;DR

This paper characterizes the structure of co-maximal subgroup graphs for various finite groups by identifying forbidden subgraphs, providing complete classifications for nilpotent and abelian groups.

Contribution

It offers new characterizations of co-maximal subgroup graphs for specific finite groups through forbidden subgraph conditions, including complete classifications for nilpotent and abelian groups.

Findings

Characterization of co-maximal subgroup graphs as cluster, triangle-free, claw-free, cograph, chordal, threshold, and split graphs.

Complete classification of co-maximal subgroup graphs for finite nilpotent groups.

Structural description of abelian groups with split co-maximal subgroup graphs.

Abstract

In this communication, the co-maximal subgroup graph $\Gamma(G)$ of a finite group $G$ is examined when $G$ is a finite nilpotent group, finite abelian group, dihedral group $D_n$, dicyclic group $Q_{2^n}$, and $p$-group. We derive the necessary and sufficient conditions for $\Gamma(G)$ to be a cluster graph, triangle-free graph, claw-free graph, cograph, chordal graph, threshold graph and split graph. For the case of finite nilpotent group, we are able to classify it entirely. Moreover, we derive the complete structure of finite abelian group $G$ such that $\Gamma(G)$ is a split graph. We leave the readers with a few unsolved questions.