In this paper we explicitly describe all the commuting pseudo inverses of a completely regular strong endomorphism of a graph from a viewpoint of combinatorics. The number of them is also given. In addition, a strong endomorphism of a graph, whose commuting pseudo inverse set coincides with its pseudo inverse set, is identified.
In this paper, split graphs with complete endomorphism-regularity are characterized explicitly. Hopefully, the main idea of the proofs can also be used for other classes of graphs.
