Commuting graph of a group action with few edges

Abstract

Let A be a group acting by automorphisms on the group G. 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 $G\setminus \left\{1\right\}$, 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.

KEYWORDS:

• Commuting graph
• friendship graph
• $\mathcal{F}$-graph
• group action

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 20D10
• 20D15
• 20D45

Acknowledgments

We are grateful to the anonymous referee for careful reading and valuable suggestions helped us to improve the article in readability and gave a chance to correct a mistake in one of the proofs.