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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.25083 |
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| _version_ | 1866911239046692864 |
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| author | Zhan, Xiongfeng Huang, Xueyi Zhou, Jin-Xin |
| author_facet | Zhan, Xiongfeng Huang, Xueyi Zhou, Jin-Xin |
| contents | This paper establishes new eigenvalue bounds for combinatorial Laplacians of simplicial complexes, extending previous results for flag complexes by Lew (2024) and general complexes by Shukla and Yogeshwaran (2020). Using elementary matrix-theoretic methods, we derive lower bounds for the eigenvalues of the combinatorial Laplacian in terms of the graph Laplacian spectrum and combinatorial parameters that measure the deviation from a flag complex. As a consequence, we obtain upper bounds on the dimension of cohomology groups. We also generalize an eigenvalue comparison inequality between a simplicial complex and its subcomplexes to arbitrary eigenvalues. As an application of the dimension bounds, we refine a result by Kahle (2007) on the vanishing of cohomology and connectivity in the neighborhood complex of the Erdős--Rényi random graph. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_25083 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Eigenvalue bounds for combinatorial Laplacians and an application to random complexes Zhan, Xiongfeng Huang, Xueyi Zhou, Jin-Xin Combinatorics 05E45 This paper establishes new eigenvalue bounds for combinatorial Laplacians of simplicial complexes, extending previous results for flag complexes by Lew (2024) and general complexes by Shukla and Yogeshwaran (2020). Using elementary matrix-theoretic methods, we derive lower bounds for the eigenvalues of the combinatorial Laplacian in terms of the graph Laplacian spectrum and combinatorial parameters that measure the deviation from a flag complex. As a consequence, we obtain upper bounds on the dimension of cohomology groups. We also generalize an eigenvalue comparison inequality between a simplicial complex and its subcomplexes to arbitrary eigenvalues. As an application of the dimension bounds, we refine a result by Kahle (2007) on the vanishing of cohomology and connectivity in the neighborhood complex of the Erdős--Rényi random graph. |
| title | Eigenvalue bounds for combinatorial Laplacians and an application to random complexes |
| topic | Combinatorics 05E45 |
| url | https://arxiv.org/abs/2510.25083 |