On The Enhanced Power Graph of a Semigroup
Sandeep Dalal, Jitender Kumar, Siddharth Singh

TL;DR
This paper explores the structure and properties of the enhanced power graph of a semigroup, providing characterizations of its connectivity, special graph types, and chromatic number, including a counterexample for uncountable chromatic number.
Contribution
It offers a comprehensive structural analysis of the enhanced power graph of semigroups, including characterizations of various graph properties and a counterexample for the chromatic number.
Findings
Characterized when the enhanced power graph is complete, bipartite, regular, a tree, or null.
Analyzed the connectedness, planarity, and degrees of the graph.
Constructed an example with uncountably infinite chromatic number.
Abstract
The enhanced power graph of a semigroup is a simple graph whose vertex set is and two vertices are adjacent if and only if for some , where is the subsemigroup generated by . In this paper, first we described the structure of for an arbitrary semigroup . Consequently, we discussed the connectedness of . Further, we characterized the semigroup such that is complete, bipartite, regular, tree and null graph, respectively. Also, we have investigated the planarity together with the minimum degree and independence number of . The chromatic number of a spanning subgraph, viz. the cyclic graph, of is proved to be countable. At the final part of this paper, we construct an example of a semigroup such…
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Taxonomy
TopicsAdvanced Graph Theory Research · Optimization and Search Problems · Graph theory and applications
