Graphs and ranks of monoids

For a finite monoid S, let (S) (nu(d)(S)) denote the least number n such that there exists a graph (directed graph) of order n with End()S. Also let rank(S) be the smallest number of elements required to generate S. In this paper, we use Cayley digraphs of monoids, to connect lower bounds of (S) ((d)(S)) to the lower bounds of rank(S). On the other hand, we connect upper bounds of rank(S) to upper bounds of nu(S) (nu(d)(S)).