Line Graph Characterization of Cyclic Subgroup Graph

TL;DR

The paper addresses when the cyclic subgroup graph $Γ(G)$ of a finite group $G$ can be realized as a line graph. It employs the forbidden-subgraph characterization of line graphs and a case-by-case group-structure analysis (via order, Sylow subgroups, and abelian/non-abelian distinctions) to narrow down to cyclic groups of prime-power order or of order $pq$ as the only possibilities. The main contribution is a complete classification: $Γ(G)$ is a line graph if and only if $G$ is cyclic of prime-power order or cyclic of order $pq$. This establishes a crisp link between the interaction of cyclic subgroups and line-graph structure, aiding structural understanding of finite groups through $Γ(G)$.

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.

Figures (1)

• Figure 1: Forbidden induced subgraphs of line graphs.

Theorems & Definitions (11)

• Lemma 2.1
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3
• proof
• Proposition 3.1
• proof
• Theorem 3.4