On The Enhanced Power Graph of a Semigroup

Abstract

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

arXiv admin note: substantial text overlap with arXiv:2107.11021