Combinatorial Courant-Fischer-Weyl Minimax Principle on Cheeger $k$-constants of Weighted Forests

Zijun Meng; Dong Zhang

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arXiv · 2510.06301

Combinatorial Courant-Fischer-Weyl Minimax Principle on Cheeger $k$-constants of Weighted Forests

Zijun Meng·Dong Zhang

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Abstract

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 $h_k=\lambda_k(\Delta_1)$ which connects the 1-Laplacian variational eigenvalues and the multiway Cheeger constants.

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BibTeX RIS

Zijun Meng, Dong Zhang. 2025-10-07. Combinatorial Courant-Fischer-Weyl Minimax Principle on Cheeger $k$-constants of Weighted Forests. https://arxiv.org/abs/2510.06301

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