Line Graph Characterization of Cyclic Subgroup Graph

Abstract

The cyclic subgroup graph ${Γ(G)}$ of a group $G$ is the simple undirected graph with cyclic subgroups as a vertex set and two distinct vertices $H_1$ and $H_2$ are adjacent if and only if $H_1 \leq H_2$ and there does not exist any cyclic subgroup $K$ such that $H_1 < K < H_2$. In this paper, we classify all the finite groups $G$ such that $Γ(G)$ is the line graph of some graph.