The cyclic graph of a semigroup

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.