On the enhanced power graph of a semigroup

Abstract

The enhanced power graph Pe(S) of a semigroup S is a simple graph whose vertex set is S and two vertices x,y∈S are adjacent if and only if x,y∈⟨z⟩ for some z∈S, where ⟨z⟩ is the subsemigroup generated by z. In this paper, first we described the structure of Pe(S) for an arbitrary semigroup S. Consequently, we discussed the connectedness of Pe(S). Further, we characterized the semigroup S such that Pe(S) is complete, bipartite, regular, tree and null graph, respectively. Also, we have investigated the planarity together with the minimum degree and independence number of Pe(S). The chromatic number of a spanning subgraph, viz. the cyclic graph, of Pe(S) is proved to be countable. At the final part of this paper, we construct an example of a semigroup S such that the chromatic number of Pe(S) need not be countable.