Intersection graph of cyclic subgroups of groups

TL;DR

This paper studies the intersection graph of cyclic subgroups of groups, classifying their structures and properties for finite groups, including planarity, girth, independence, and regularity.

Contribution

It provides a comprehensive classification of the intersection graphs of cyclic subgroups for finite groups, including structural, planarity, and numerical properties.

Findings

Classified finite groups with specific intersection graph structures.

Determined girth values for intersection graphs of cyclic subgroups.

Analyzed planarity, independence, and regularity of these graphs.

Abstract

Let $G$ be a group. The intersection graph of cyclic subgroups of $G$, denoted by $\mathscr I_c(G)$, is a graph having all the proper cyclic subgroups of $G$ as its vertices and two distinct vertices in $\mathscr I_c(G)$ are adjacent if and only if their intersection is non-trivial. In this paper, we classify the finite groups whose intersection graph of cyclic subgroups is one of totally disconnected, complete, star, path, cycle. We show that for a given finite group $G$, $girth(\mathscr I_c (G)) \in \{3, \infty\}$. Moreover, we classify all finite non-cyclic abelian groups whose intersection graph of cyclic subgroups is planar. Also for any group $G$, we determine the independence number, clique cover number of $\mathscr I_c (G)$ and show that $\mathscr I_c (G)$ is weakly $\alpha$-perfect. Among the other results, we determine the values of $n$ for which $\mathscr I_c (\mathbb{Z}_n)$ is regular and estimate its domination number.