Spectral bounds for vertex-weighted Laplacians of simplicial complexes
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918243460972544 |
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| author | Han, Yueli Lu, Lu |
| author_facet | Han, Yueli Lu, Lu |
| contents | The vertex-weighted Laplacian naturally extends the combinatorial Laplacian for simplicial complexes. Inspired by Lew's foundational techniques for vertex-weighted Laplacians, we present a comprehensive spectral analysis of this operator. First, we determine how basic operations, including joins, complements, and Alexander duals, affect its spectrum. This yields a sharp upper bound on the spectral radius in terms of vertex weights, along with a lower bound on the multiplicity at which this bound is attained. Second, we establish a sharp lower bound for the spectral gap and characterize when the equality holds. Third, explicit lower bounds for the remaining eigenvalues are derived, linking the vertex-weighted Laplacian spectrum to that of a related weighted graph. Finally, we reveal new spectral relations between a simplicial complex and its subcomplexes. These results not only generalize numerous known theorems on combinatorial Laplacians but also provide deeper spectral insights into simplicial structures, ultimately unifying and extending a broad range of earlier work in this field. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_10364 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral bounds for vertex-weighted Laplacians of simplicial complexes Han, Yueli Lu, Lu Combinatorics 05C50, 05C65, 05E45 The vertex-weighted Laplacian naturally extends the combinatorial Laplacian for simplicial complexes. Inspired by Lew's foundational techniques for vertex-weighted Laplacians, we present a comprehensive spectral analysis of this operator. First, we determine how basic operations, including joins, complements, and Alexander duals, affect its spectrum. This yields a sharp upper bound on the spectral radius in terms of vertex weights, along with a lower bound on the multiplicity at which this bound is attained. Second, we establish a sharp lower bound for the spectral gap and characterize when the equality holds. Third, explicit lower bounds for the remaining eigenvalues are derived, linking the vertex-weighted Laplacian spectrum to that of a related weighted graph. Finally, we reveal new spectral relations between a simplicial complex and its subcomplexes. These results not only generalize numerous known theorems on combinatorial Laplacians but also provide deeper spectral insights into simplicial structures, ultimately unifying and extending a broad range of earlier work in this field. |
| title | Spectral bounds for vertex-weighted Laplacians of simplicial complexes |
| topic | Combinatorics 05C50, 05C65, 05E45 |
| url | https://arxiv.org/abs/2512.10364 |