Commuting graph of a group action with few edges

2022/07/07 by Güloğlu, İsmail Ş., Ercan, Gülin
#20D10 #20D15 #20D45 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2207.03193

Abstract

Let A be a group acting by automorphisms on the group G. The commuting graph Γ(G,A) of A-orbits of this action is the simple graph with vertex set \x^A : 1≠ x ∈ G \, the set of all A-orbits on G∖ \1\, where two distinct vertices x^A and y^A are joined by an edge if and only if there exist x₁∈ x^A and y₁∈ y^A such that [x₁,y₁]=1. The present paper characterizes the groups G for which Γ(G,A) is an F-graph, that is, a connected graph which contains at most one vertex whose degree is not less than three.

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