2021/07/25 by Sandeep Dalal, Jitender Kumar, Dalal, Sandeep +3
Computer Science · Mathematics · #Advanced Graph Theory Research #FOS: Mathematics #Graph theory and applications #Group Theory (math.GR) #Optimization and Search Problems
paper · pdf · doi:10.48550/arxiv.2107.11793
openalex publication_date 2021/07/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The enhanced power graph \mathcal 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 \mathcal Pe(S) for an arbitrary semigroup S. Consequently, we discussed the connectedness of \mathcal Pe(S). Further, we characterized the semigroup S such that \mathcal 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 \mathcal Pe(S). The chromatic number of a spanning subgraph, viz. the cyclic graph, of \mathcal 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 \mathcal Pe(S) need not be countable.