  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.
