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

The cyclic graph gamma(S) of a semigroup S is the simple undirected 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 determine the clique number of gamma(S) for an arbitrary semigroup S. Further, 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.