In this paper, we construct nilpotent semigroups S such that Sn = {0}, Sn−1 6≠{0} and Γ(S) is a refinement of the star graph K1,n−3 with center c together with finitely many or infinitely many end vertices adjacent to c, for each finite positive integer n ≥ 5. We also give counting formulae to calculate the number of the mutually non-isomorphic nilpo- tent semigroups when n = 5, 6 and in finite cases.