Cayley graphs and power digraphs over completely simple semigroups

Let S_1 and S_2 be completely simple semigroups. By investigating the intrinsic relationship between Cayley graphs and directed power graphs, we obtain a necessary and sufficient condition that the power digraph of S_1 is isomorphic to Cayley graph of S_2 . Moreover, we provide an exact characterization of the completely simple semigroup whose power digraph is isomorphic to Cayley graph of some completely simple semigroup and reveal the structure of the completely simple semigroup whose Cayley graph is isomorphic to the power digraph of some completely simple semigroup.