Communications in Algebra, vol.51, no.7, pp.2791-2810, 2023 (SCI-Expanded)
Let A be a group acting by automorphisms on the group G. The commuting graph (Formula presented.) of A-orbits of this action is the simple graph with vertex set (Formula presented.), the set of all A-orbits on (Formula presented.), where two distinct vertices xA and yA are joined by an edge if and only if there exist (Formula presented.) and (Formula presented.) such that (Formula presented.). The present paper characterizes the groups G for which (Formula presented.) is an (Formula presented.) -graph, that is, a connected graph which contains at most one vertex whose degree is not less than three.