Let G be a refinement of a star graph with center c. Let G*c be the subgraph of G inducedon the vertex set V.G\{c or end vertices adjacent to cg. In this paper, we completelydetermine the structure of commutative zero-divisor semigroups S whose zero-divisorgraph G=Г(S) and S satisfy one of the following properties: (1) G*c has at least twoconnected components, (2) G*c is a cycle graph Cn of length n≥5, (3) G*c is a path graphPn with n≥ 6, (4) S is nilpotent andГ(S) is a finite or an infinite star graph. For any finiteor infinite cardinal number n2, we prove that for any nilpotent semigroup S with zeroelement 0, S4=0 if Г(S) is a star graph K1m n. We prove that there is exactly one nilpotentsemigroup S such that S3≠0 and Г(S)≡K1;n. For several classes of finite graphs G which are refinements of a star graph, we also obtain formulas to count the number ofnon-isomorphic corresponding semigroups.