Equality of various graphs on finite semigroups

TL;DR

This paper classifies finite semigroups based on when different graphs associated with them, like power, cyclic, enhanced power, and commuting graphs, are equal, extending known group results.

Contribution

It provides a comprehensive classification of finite semigroups where various graphs coincide, generalizing previous group-based findings.

Findings

Characterization of semigroups with equal power and cyclic graphs

Conditions for the completeness of each graph type

Generalization of group results to semigroups

Abstract

In this paper, we consider various graphs, namely: power graph, cyclic graph, enhanced power graph and commuting graph, on a finite semigroup $S$. For an arbitrary pair of these four graphs, we classify finite semigroups such that the graphs in this pair are equal. In this connection, for each of the graph we also give a necessary and sufficient condition on $S$ such that it is complete. The work of this paper generalize the corresponding results obtained for groups.