Abstract
Let A be a group acting by automorphisms on the group G. The commuting graph of A-orbits of this action is the simple graph with vertex set
, the set of all A-orbits on
, where two distinct vertices xA and yA are joined by an edge if and only if there exist
and
such that
. The present paper characterizes the groups G for which
is an
-graph, that is, a connected graph which contains at most one vertex whose degree is not less than three.
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.