On the deep commuting graph of a finite group

TL;DR

This paper studies the deep commuting graph of finite groups, revealing conditions for completeness, classifying groups with perfect graphs, and exploring various properties and special cases.

Contribution

It provides new characterizations of the deep commuting graph, including conditions for completeness, perfection, and equivalence with other graphs for specific classes of groups.

Findings

Deep commuting graph is complete iff the group is cyclic.

Classified finite simple, symmetric, and alternating groups with perfect deep commuting graphs.

Analyzed properties like Eulerianess, universality, and connectedness of the graphs.

Abstract

Let $G$ be a finite group and let $\tilde{G}$ be a Schur cover of $G$. The deep commuting graph $\Delta_D(G)$ of $G$ is a simple graph with vertex set $G$, where two distinct vertices are adjacent if their pre-images commute in $\tilde{G}$. The deep commuting graph of a finite group was first introduced in [P. J. Cameron and B. Kuzma, Between the enhanced power graph and the commuting graph, {\it J. Graph Theory} {\bf 102} (2023), no. 2, 295--303], where the authors have shown that $\Delta_D(G)$ is fixed irrespective of the choice of the Schur cover $\tilde{G}$. In this paper, we first prove that $\Delta_D(G)$ is complete if and only if $G$ is cyclic. Also, we classify finite simple groups, symmetric groups and alternating groups, for which $\Delta_D(G)$ is perfect. In addition, explore several other properties of $\Delta_D(G)$ like Eulerianess, universality and connectedness of reduced deep commuting graphs. Next, we classify the finite abelian groups for which deep commuting graphs coincide with enhance power graphs. We also characterize the dominant vertices for the deep commuting graphs of finite abelian groups and examine the connectedness of the associated reduced deep commuting graphs. These properties of the deep commuting graphs for the non abelian groups like symmetric groups, alternating groups, dihedral groups, generalized quaternion group and Heisenberg groups are also discussed.