Combinatorial Laplacians and Relative Homology of Complex Pairs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909733213962240 |
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| author | Zhan, Xiongfeng Huang, Xueyi Lu, Lu |
| author_facet | Zhan, Xiongfeng Huang, Xueyi Lu, Lu |
| contents | As a discretization of the Hodge Laplacian, the combinatorial Laplacian of simplicial complexes has garnered significant attention. In this paper, we study combinatorial Laplacians for complex pairs $(X, A)$, where $A$ is a subcomplex of a simplicial complex $X$. We establish a relative version of the matrix-tree theorem for complex pairs, which generalizes both the matrix-tree theorem for simplicial complexes proved by Duval, Klivans, and Martin (2009) and the result for Dirichlet eigenvalues of graph pairs by Chung (1996). Furthermore, we derive several lower bounds for the spectral gaps of complex pairs and characterize the equality case for one sharp lower bound. As by-products, we obtain sufficient conditions for the vanishing of relative homology. Our results demonstrate that the combinatorial Laplacians for complex pairs are closely related to relative homology. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_16381 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Combinatorial Laplacians and Relative Homology of Complex Pairs Zhan, Xiongfeng Huang, Xueyi Lu, Lu Combinatorics Algebraic Topology 05E45, 55U10 As a discretization of the Hodge Laplacian, the combinatorial Laplacian of simplicial complexes has garnered significant attention. In this paper, we study combinatorial Laplacians for complex pairs $(X, A)$, where $A$ is a subcomplex of a simplicial complex $X$. We establish a relative version of the matrix-tree theorem for complex pairs, which generalizes both the matrix-tree theorem for simplicial complexes proved by Duval, Klivans, and Martin (2009) and the result for Dirichlet eigenvalues of graph pairs by Chung (1996). Furthermore, we derive several lower bounds for the spectral gaps of complex pairs and characterize the equality case for one sharp lower bound. As by-products, we obtain sufficient conditions for the vanishing of relative homology. Our results demonstrate that the combinatorial Laplacians for complex pairs are closely related to relative homology. |
| title | Combinatorial Laplacians and Relative Homology of Complex Pairs |
| topic | Combinatorics Algebraic Topology 05E45, 55U10 |
| url | https://arxiv.org/abs/2507.16381 |