Zero-divisor semigroups and refinements of a star graph

Let GG be a refinement of a star graph with center cc. Let Gc∗ be the subgraph of GG induced on the vertex set V(G)∖{c or end vertices adjacent to c}V(G)∖{c or end vertices adjacent to c}. In this paper, we completely determine the structure of commutative zero-divisor semigroups SS whose zero-divisor graph G=Γ(S)G=Γ(S) and SS satisfy one of the following properties: (1) Gc∗ has at least two connected components, (2) Gc∗ is a cycle graph CnCn of length n≥5n≥5, (3) Gc∗ is a path graph PnPn with n≥6n≥6, (4) SS is nilpotent and Γ(S)Γ(S) is a finite or an infinite star graph. For any finite or infinite cardinal number n≥2n≥2, we prove that for any nilpotent semigroup SS with zero element 0, S4=0S4=0 if Γ(S)Γ(S) is a star graph K1,nK1,n. We prove that there is exactly one nilpotent semigroup SS such that S3≠0S3≠0 and Γ(S)≅K1,nΓ(S)≅K1,n. For several classes of finite graphs GG which are refinements of a star graph, we also obtain formulas to count the number of non-isomorphic corresponding semigroups.