Zero-divisor semigroups and refinements of a star graph

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