Commuting graph of a group action with few edges

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.

22 pages