Zero-divisor semigroups and refinements of a star graph |
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Authors: | Tongsuo Wu Qiong Liu |
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Institution: | Department of Mathematics, Shanghai Jiaotong University, Shanghai 200240, PR China |
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Abstract: | Let G be a refinement of a star graph with center c. Let 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) has at least two connected components, (2) is a cycle graph Cn of length n≥5, (3) is a path graph Pn 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, S4=0 if Γ(S) is a star graph K1,n. We prove that there is exactly one nilpotent semigroup S such that S3≠0 and Γ(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 of non-isomorphic corresponding semigroups. |
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Keywords: | Refinements of star graphs Semigroups Nilpotent semigroups Counting formulas |
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