In this paper, we construct nilpotent semigroups S such that $S^n={0}$, $S^{n-1}{ eq}{0}$ and ${Gamma}(S)$ is a refinement of the star graph $K_{1,n-3}$ with center $c$ together with finitely many or infinitely many end vertices adjacent to $c$, for each finite positive integer $n{geq}5$. We also give counting formulae to calculate the number of the mutually non-isomorphic nilpotent semigroups when $n=5$, 6 and in finite cases.