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

Zijun Meng, Dong Zhang

TL;DR

The paper develops the first combinatorial analogue of the Courant-Fischer-Weyl minimax principle for Cheeger $k$-constants on weighted forests, providing max-min and min-max reformulations via subpartitions and union-closed families. It proves a nonlinear, $L^1$-Rayleigh-quotient version of the minimax principle and shows that, on forests, the Cheeger constants $h_k(G)$ coincide with the $k$-th minimax eigenvalues of the forest $1$-Laplacian $\Delta_1$ and are independent of the admissible index. The authors further derive a refined multi-way Cheeger inequality using the graph $p$-Laplacian and a loop-count parameter $\beta$, obtaining bounds that tighten as $p\to1$ and extend to graphs with few cycles via $h_{k-\beta}(G)\le \lambda_k(\Delta_1)\le h_k(G)$. Collectively, the work unifies combinatorial and spectral viewpoints, enabling sharper isoperimetric estimates, robust eigenvalue characterizations for forests, and extensions to graphs near-forests with controlled cycle structure.

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 $h_k=λ_k(Δ_1)$ which connects the 1-Laplacian variational eigenvalues and the multiway Cheeger constants.

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

TL;DR

The paper develops the first combinatorial analogue of the Courant-Fischer-Weyl minimax principle for Cheeger -constants on weighted forests, providing max-min and min-max reformulations via subpartitions and union-closed families. It proves a nonlinear, -Rayleigh-quotient version of the minimax principle and shows that, on forests, the Cheeger constants coincide with the -th minimax eigenvalues of the forest -Laplacian and are independent of the admissible index. The authors further derive a refined multi-way Cheeger inequality using the graph -Laplacian and a loop-count parameter , obtaining bounds that tighten as and extend to graphs with few cycles via . Collectively, the work unifies combinatorial and spectral viewpoints, enabling sharper isoperimetric estimates, robust eigenvalue characterizations for forests, and extensions to graphs near-forests with controlled cycle structure.

Abstract

We establish novel max-min and minimax characterizations of Cheeger -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 -Laplacian eigenvalues; and we present a combinatorial proof for the equality which connects the 1-Laplacian variational eigenvalues and the multiway Cheeger constants.

Paper Structure

This paper contains 15 sections, 19 theorems, 79 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Higher Order Cheeger Inequalities
  3. Minimax Principle
  4. Main Results
  5. Applications on min-max eigenvalues of Graph 1-Laplacian of Forests
  6. Applications on Refined Multi-way Cheeger Inequalities
  7. Proofs
  8. Proofs of the Main Theorems (Combinatorial)
  9. Proof of Proposition \ref{['prop:unioninv']}
  10. Proof of Theorem \ref{['thm:h_k=H_k']}
  11. Proofs of Theorem \ref{['thm:cycle-number-inequ']} and Theorem \ref{['thm:CombinatorialCourant-Fischer']}
  12. Proofs of the Main Applications (Analytical)
  13. Tools and Properties of Graph $p$-Laplacian
  14. Proofs of Theorem \ref{['thm:1-Lap=h']} and Theorem \ref{['thm:nonlinearCourant-Fischer']}
  15. Proofs of Other Results in Sections \ref{['sec:minmax-eigen-1-Lap']} and \ref{['sec:refined-Cheeger']}

Key Result

Theorem 1

For any forest $G=(V,E,\mu,w)$, we have where denotes the union-closed family generated by the subpartition $(A_1,\dots,A_{k})$ of $V$.

Figures (2)

  • Figure 1: Logic Flow
  • Figure 2: Choosing $v_1,\dots,v_{k-1}$.

Theorems & Definitions (38)

  • Theorem 1: combinatorial Courant-Fischer-Weyl minimax principle
  • Theorem 2: nonlinear Courant-Fischer-Weyl minimax principle
  • Proposition 1
  • Theorem 3
  • Theorem 4
  • Definition 1: admissible index
  • Theorem 5
  • Corollary 1
  • Corollary 2: Deidda-thesisDeidda
  • Corollary 3
  • ...and 28 more