On bipartite zero-divisor graphs |
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Authors: | Dancheng Lu Tongsuo Wu |
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Affiliation: | a Department of Mathematics, Suzhou University, Suzhou 215006, PR China b Department of Mathematics, Shanghai Jiaotong University, Shanghai 200240, PR China |
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Abstract: | A (finite or infinite) complete bipartite graph together with some end vertices all adjacent to a common vertex is called a complete bipartite graph with a horn. For any bipartite graph G, we show that G is the graph of a commutative semigroup with 0 if and only if it is one of the following graphs: star graph, two-star graph, complete bipartite graph, complete bipartite graph with a horn. We also prove that a zero-divisor graph is bipartite if and only if it contains no triangles. In addition, we give all corresponding zero-divisor semigroups of a class of complete bipartite graphs with a horn and determine which complete r-partite graphs with a horn have a corresponding semigroup for r≥3. |
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Keywords: | Zero-divisor semigroup Zero-divisor graph Bipartite graph Full vertex A complete bipartite graph with a horn |
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