Signless Laplacian spectral radius of simplicial complexes without holes
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866914524737568768 |
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| author | Fan, Yi-Zheng She, Chuan-Ming Zhang, Huan-Zhi |
| author_facet | Fan, Yi-Zheng She, Chuan-Ming Zhang, Huan-Zhi |
| contents | We study a spectral analog of the Turán problem for simplicial complexes. Specifically, we consider the extremal problem of maximizing the signless Laplacian spectral radius among simplicial complexes without holes. We determine the structure of the simplicial complex attaining the maximum spectral radius, extending classical extremal results for graphs without cycles to the setting of higher-dimensional simplicial complexes. More generally, we establish an upper bound on the signless Laplacian spectral radius of simplicial complexes with prescribed Betti numbers. As an application, using the connection between the signless Laplacian spectral radius and the face numbers of a simplicial complex, we derive bounds on Turán numbers for both hypergraphs and simplicial complexes. Our technique involves the canonical Alexander dual of perfect matchings and coloring of simplicial complexes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_22518 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Signless Laplacian spectral radius of simplicial complexes without holes Fan, Yi-Zheng She, Chuan-Ming Zhang, Huan-Zhi Combinatorics 05C35, 05E45, 05C65, 55U05 We study a spectral analog of the Turán problem for simplicial complexes. Specifically, we consider the extremal problem of maximizing the signless Laplacian spectral radius among simplicial complexes without holes. We determine the structure of the simplicial complex attaining the maximum spectral radius, extending classical extremal results for graphs without cycles to the setting of higher-dimensional simplicial complexes. More generally, we establish an upper bound on the signless Laplacian spectral radius of simplicial complexes with prescribed Betti numbers. As an application, using the connection between the signless Laplacian spectral radius and the face numbers of a simplicial complex, we derive bounds on Turán numbers for both hypergraphs and simplicial complexes. Our technique involves the canonical Alexander dual of perfect matchings and coloring of simplicial complexes. |
| title | Signless Laplacian spectral radius of simplicial complexes without holes |
| topic | Combinatorics 05C35, 05E45, 05C65, 55U05 |
| url | https://arxiv.org/abs/2507.22518 |