Which coprime graphs are divisor graphs?

TL;DR

This work investigates when the coprime graph $\Gamma(G)$ of a finite group $G$ is a divisor graph. By establishing that $\Gamma(G)$ is a generalized lexicographic product and leveraging transitive orientations, the authors reduce the problem to analyzing auxiliary graphs $\mathcal{O}_G$ and $\mathcal{N}_G$, yielding broad structural criteria. They prove that several classical graphs on groups (power, reduced power, and order graphs) are divisor graphs, and provide complete classifications for $|\pi(G)|\le 4$ and for key families: dihedral and generalized quaternion groups, symmetric and alternating groups, direct products, and sporadic simple groups. The results connect prime-divisor structure and element-order sets to the divisor-graph property, offering precise conditions and complete lists in many cases, with notable implications for CP-groups and finite group theory graph classifications.

Abstract

For a finite group $G$, the coprime graph $Γ(G)$ of $G$ is a graph with vertex set $G$, in which two distinct vertices $a$ and $b$ are adjacent if the order of $a$ and the order of $b$ are coprime. In this paper, we first give a characterization for which generalized lexicographic products are divisor graphs. As applications, we show that every of power graph, reduced power graph and order graph is a divisor graph, which also implies the main result in [N. Takshak, A. Sehgal, A. Malik, Power graph of a finite group is always divisor graph, Asian-Eur. J. Math. 16 (2023), ID: 2250236]. Then, we prove that the coprime graph of a group is a generalized lexicographic product, and give two characterizations for which coprime graphs are divisor graphs. We also describe the groups $G$ with $|π(G)|\le 4$, whose coprime graph is a divisor graph. Finally, we classify the finite groups $G$ so that $Γ(G)$ is a divisor graph if $G$ is a nilpotent group, a dihedral group, a generalized quaternion group, a symmetric group, an alternating group, a direct product of two non-trivial groups, and a sporadic simple group.

Figures (7)

• Figure 1: A block graph which is not a divisor graph
• Figure 2: Two possibilities for $\mathcal{O}_G$
• Figure 3: Two possible transitive orientations for $\mathcal{O}_G$
• Figure 4: A transitive orientation for $\mathcal{O}_G$
• Figure 5: $\mathcal{N}_G$ with $V(\mathcal{N}_G)=\{p,q,r,s,pq,pr,rs,qs\}$

Theorems & Definitions (29)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Theorem 2.6
• Theorem 3.1
• Corollary 3.2
• Corollary 3.3
• Corollary 3.4