The cyclic graph of a semigroup

TL;DR

This paper studies the cyclic graph of a semigroup, classifies semigroups based on graph properties, and determines key graph invariants like clique and independence numbers.

Contribution

It provides classifications of semigroups with specific cyclic graph structures and calculates important graph parameters for various classes of semigroups.

Findings

Classified semigroups with complete, bipartite, tree, regular, null cyclic graphs.

Determined the clique number for any semigroup's cyclic graph.

Established bounds and characterized semigroups for the independence number in finite monogenic cases.

Abstract

The cyclic graph $\Gamma(S)$ of a semigroup $S$ is the simple graph whose vertex set is $S$ and two vertices $x, y$ are adjacent if the subsemigroup generated by $x$ and $y$ is monogenic. In this paper, we classify the semigroup $S$ such that whose cyclic graph $\Gamma(S)$ is complete, bipartite, tree, regular and a null graph, respectively. Further, we determine the clique number of $\Gamma(S)$ for an arbitrary semigroup $S$. We obtain the independence number of $\Gamma(S)$ if $S$ is a finite monogenic semigroup. At the final part of this paper, we give bounds for independence number of $\Gamma(S)$ if $S$ is a semigroup of bounded exponent and we also characterize the semigroups attaining the bounds.