Intersection subgroup graph with forbidden subgraphs

TL;DR

This paper studies the intersection subgroup graph of groups, classifying groups based on whether their graphs belong to certain graph classes, and provides complete classifications for specific types of groups.

Contribution

It characterizes groups whose intersection subgroup graphs are in various graph classes and classifies simple Lie type groups with cograph intersection graphs.

Findings

Nilpotent groups with certain graph class properties identified

Simple Lie type groups classified by intersection subgroup graph type

Torsion-free nilpotent groups have graphs that are neither cographs nor chordal

Abstract

Let $G$ be a group. The intersection subgroup graph of $G$ (introduced by Anderson et al. \cite{anderson}) is the simple graph $\Gamma_{S}(G)$ whose vertices are those non-trivial subgroups say $H$ of $G$ with $H\cap K=\{e\}$ for some non-trivial subgroup $K$ of $G$; two distinct vertices $H$ and $K$ are adjacent if and only if $H\cap K=\{e\}$, where $e$ is the identity element of $G$. In this communication, we explore the groups whose intersection subgroup graph belongs to several significant graph classes including cluster graphs, perfect graphs, cographs, chordal graphs, bipartite graphs, triangle-free and claw-fee graphs. We categorize each nilpotent group $G$ so that $\Gamma_S(G)$ belongs to the above classes. We entirely classify the simple group of Lie type whose intersection subgroup graph is a cograph. Moreover, we deduce that $\Gamma_{S}(G)$ is neither a cograph nor a chordal graph if $G$ is a torsion-free nilpotent group.