Commuting graph of a group action with few edges

TL;DR

This paper characterizes groups based on the structure of their commuting graphs under group actions, specifically identifying when these graphs are connected with at most one high-degree vertex.

Contribution

It provides a classification of groups for which the commuting graph of A-orbits is an -graph, a connected graph with at most one vertex of degree three or higher.

Findings

Identifies conditions for -graph structure in commuting graphs

Characterizes groups with such commuting graph properties

Provides a framework for analyzing group actions via graph theory

Abstract

Let $A$ be a group acting by automorphisms on the group $G.$ \textit{The commuting graph $\Gamma(G,A)$ of $A$-orbits} of this action is the simple graph with vertex set $\{x^{A} : 1\ne x \in G \}$, the set of all $A$-orbits on $G\setminus \{1\}$, where two distinct vertices $x^{A}$ and $y^{A}$ are joined by an edge if and only if there exist $x_{1}\in x^{A}$ and $y_{1}\in y^{A}$ such that $[x_{1},y_{1}]=1$. The present paper characterizes the groups $G$ for which $\Gamma(G,A)$ is an $\mathcal{F}$-graph, that is, a connected graph which contains at most one vertex whose degree is not less than three.