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Enhanced power graphs of finite groups with cograph structure

Daniela Bubboloni, Francesco Fumagalli, Cheryl E. Praeger

TL;DR

The paper develops a systematic theory of enhanced power graphs $\mathcal{E}(G)$ for finite groups, proving that a $\mathcal{E}(G)$ being a cograph automatically implies it is chordal and further identifying when it is quasi-threshold or a block graph. It establishes a practical group-theoretic criterion (based on maximal cyclic subgroups) for $\mathcal{E}(G)$ to be a cograph, and derives a nilpotent classification $G= P\times C_n$ with $(p,n)=1$ that characterises when $\mathcal{E}(G)$ is cograph/chordal/$C_4$-free. The work then classifies finite nonabelian simple groups with $\mathcal{E}(G)$ a cograph (Theorem B) and provides detailed conditions for $C_4$-free instances, relying on CFSG, with explicit families such as $\mathrm{PSL}_2(q)$, $\mathrm{Sz}(q)$, and $\mathrm{PSL}_3(4)$. Finally, it connects these graph-theoretic classifications to block-graph structures, partitions of cyclic subgroups, and contrasts with power-graph results, outlining several open problems. These results advance understanding of how cyclic subgroups interact in finite groups via graph-theoretic properties, offering efficient structural characterisations and a framework for future exploration.

Abstract

The enhanced power graph, $\mathcal{E}(G)$, of a group $G$ has vertex set $G$ and two elements are adjacent if they generate a cyclic subgroup. In the case of finite groups, we identify some striking and unexpected properties of these graphs, as well as links between properties of $\mathcal{E}(G)$ and properties of the group $G$. We prove that if $\mathcal{E}(G)$ is a cograph then it is also a chordal graph. Making use of properties of simplicial vertices, we characterise the finite groups $G$ whose enhanced power graph is diamond-free or a block graph. We also characterise the finite groups having enhanced power graph a cograph or a quasi-threshold graph, and those with $C_4$-free enhanced power graph. We use these characterisations to classify the finite nonabelian simple groups whose enhanced power graph is a cograph and give information on the finite simple groups whose enhanced power graph is $C_4$-free. Some open problems are posed.

Enhanced power graphs of finite groups with cograph structure

TL;DR

The paper develops a systematic theory of enhanced power graphs for finite groups, proving that a being a cograph automatically implies it is chordal and further identifying when it is quasi-threshold or a block graph. It establishes a practical group-theoretic criterion (based on maximal cyclic subgroups) for to be a cograph, and derives a nilpotent classification with that characterises when is cograph/chordal/-free. The work then classifies finite nonabelian simple groups with a cograph (Theorem B) and provides detailed conditions for -free instances, relying on CFSG, with explicit families such as , , and . Finally, it connects these graph-theoretic classifications to block-graph structures, partitions of cyclic subgroups, and contrasts with power-graph results, outlining several open problems. These results advance understanding of how cyclic subgroups interact in finite groups via graph-theoretic properties, offering efficient structural characterisations and a framework for future exploration.

Abstract

The enhanced power graph, , of a group has vertex set and two elements are adjacent if they generate a cyclic subgroup. In the case of finite groups, we identify some striking and unexpected properties of these graphs, as well as links between properties of and properties of the group . We prove that if is a cograph then it is also a chordal graph. Making use of properties of simplicial vertices, we characterise the finite groups whose enhanced power graph is diamond-free or a block graph. We also characterise the finite groups having enhanced power graph a cograph or a quasi-threshold graph, and those with -free enhanced power graph. We use these characterisations to classify the finite nonabelian simple groups whose enhanced power graph is a cograph and give information on the finite simple groups whose enhanced power graph is -free. Some open problems are posed.
Paper Structure (13 sections, 35 theorems, 33 equations, 1 table)

This paper contains 13 sections, 35 theorems, 33 equations, 1 table.

Table of Contents

  1. Introduction and main results
  2. Basic graph theoretical notions
  3. power graph and enhanced power graph
  4. Simplicial vertices in enhanced power graphs.
  5. The role of the cycliciser for the enhanced power graph.
  6. Chordal and cograph properties of enhanced power graphs
  7. Quasi-threshold graphs and the group theoretical characterisation of groups whose enhanced power graph is a cograph.
  8. Enhanced power graphs of nilpotent groups
  9. Induced paths and cycles in enhanced power graphs
  10. $C_4$-free enhanced power graphs.
  11. enhanced power graphs which are block graphs
  12. enhanced power graphs of finite nonabelian simple groups
  13. Some comments on power graphs

Key Result

Lemma 2.4

Suppose that a graph $\Gamma$ contains either an induced path $\gamma\cong P_n$, with $n\ge3$, or an induced cycle $\mathcal{C}\cong C_n$, with $n\geq 4$. Then the following hold:

Theorems & Definitions (77)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • proof
  • Lemma 2.4
  • proof
  • Definition 3.1
  • Lemma 3.2
  • Corollary 3.3
  • Lemma 3.4
  • ...and 67 more