On the clique number and independence number of the cyclic graph of a semigroup

Abstract

The cyclic graph [Formula: see text] of a semigroup [Formula: see text] is the simple undirected graph whose vertex set is [Formula: see text] and two vertices [Formula: see text] are adjacent if the subsemigroup generated by [Formula: see text] and [Formula: see text] is monogenic. In this paper, we determine the clique number of [Formula: see text] for an arbitrary semigroup [Formula: see text]. Further, we obtain the independence number of [Formula: see text] if [Formula: see text] is a finite monogenic semigroup. At the final part of this paper, we give bounds for independence number of [Formula: see text] if [Formula: see text] is a semigroup of bounded exponent and we also characterize the semigroups attaining the bounds.