Combinatorial Courant-Fischer-Weyl Minimax Principle on Cheeger $k$-constants of Weighted Forests
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914080166510592 |
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| author | Meng, Zijun Zhang, Dong |
| author_facet | Meng, Zijun Zhang, Dong |
| contents | We establish novel max-min and minimax characterizations of Cheeger $k$-constants in weighted forests, thereby providing the first combinatorial analogue of the Courant-Fischer-Weyl minimax principle. As for applications, we prove that the forest 1-Laplacian variational eigenvalues are independent of the choice of typical indexes; we propose a refined higher order Cheeger inequality involving numbers of loops of graphs and $p$-Laplacian eigenvalues; and we present a combinatorial proof for the equality $h_k=λ_k(Δ_1)$ which connects the 1-Laplacian variational eigenvalues and the multiway Cheeger constants. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_06301 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Combinatorial Courant-Fischer-Weyl Minimax Principle on Cheeger $k$-constants of Weighted Forests Meng, Zijun Zhang, Dong Combinatorics Spectral Theory We establish novel max-min and minimax characterizations of Cheeger $k$-constants in weighted forests, thereby providing the first combinatorial analogue of the Courant-Fischer-Weyl minimax principle. As for applications, we prove that the forest 1-Laplacian variational eigenvalues are independent of the choice of typical indexes; we propose a refined higher order Cheeger inequality involving numbers of loops of graphs and $p$-Laplacian eigenvalues; and we present a combinatorial proof for the equality $h_k=λ_k(Δ_1)$ which connects the 1-Laplacian variational eigenvalues and the multiway Cheeger constants. |
| title | Combinatorial Courant-Fischer-Weyl Minimax Principle on Cheeger $k$-constants of Weighted Forests |
| topic | Combinatorics Spectral Theory |
| url | https://arxiv.org/abs/2510.06301 |