Permutability graph of cyclic subgroups

TL;DR

This paper classifies finite groups based on properties of their permutability graphs of cyclic subgroups, exploring structural, planarity, and connectivity aspects of these graphs.

Contribution

It provides a comprehensive classification of finite groups whose permutability graphs of cyclic subgroups exhibit specific graph-theoretic properties and analyzes their structural characteristics.

Findings

Classified groups with bipartite, star, triangle-free, and other graph properties.

Determined conditions for planarity of abelian groups' permutability graphs.

Analyzed connectedness, diameter, girth, and regularity of these graphs.

Abstract

Let $G$ be a group. \textit{The permutability graph of cyclic subgroups of $G$}, denoted by $\Gamma_c(G)$, is a graph with all the proper cyclic subgroups of $G$ as its vertices and two distinct vertices in $\Gamma_c(G)$ are adjacent if and only if the corresponding subgroups permute in $G$. In this paper, we classify the finite groups whose permutability graph of cyclic subgroups belongs to one of the following: bipartite, tree, star graph, triangle-free, complete bipartite, $P_n$, $C_n$, $K_4$, $K_{1,3}$-free, unicyclic. We classify abelian groups whose permutability graph of cyclic subgroups are planar. Also we investigate the connectedness, diameter, girth, totally disconnectedness, completeness and regularity of these graphs.