Cayley graphs and directed power graph of completely simple semigroup

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