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&NOTEQUexpressionL;x is an element of 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 is an element of x(A) and y1 is an element of y(A) such that [x(1), y(1)] = 1. The present paper characterizes the groups G for which gamma(G,A) is an F-graph, that is, a connected graph which contains at most one vertex whose degree is not less than three.