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Cheeger inequalities on simplicial complexes

Jürgen Jost, Dong Zhang

TL;DR

The paper addresses extending Cheeger inequalities to higher-dimensional simplicial complexes via the Eckmann up-Laplacian. By identifying a suitable Cheeger constant and employing a synthesis of simplicial topology, signed-graph theory, Gromov filling, and nonlinear p-Laplacian dualities, it proves a two-sided bound relating the first nonzero eigenvalue to a Cheeger constant for typical uniform triangulations of $(d+1)$-manifolds. It also develops a complementary bound from the top eigenvalue $d+2$ using a signed-graph reduction and introduces a novel Cheeger constant based on generalized multisets to bound the ground-state eigenvalue of the up $p$-Laplacian, with full p-Laplacian generality. These results provide a rigorous higher-dimensional Cheeger theory, linking geometry, topology, and nonlinear spectral analysis, and offering a framework for high-dimensional expander analysis and manifold triangulations. The approach combines graph-to-manifold approximation, nonlinear spectral duality, and a robust $p$-Laplacian theory to yield universal-type inequalities that mirror the classical graph and manifold Cheeger inequalities in a higher-dimensional setting.

Abstract

Cheeger-type inequalities in which the decomposability of a graph and the spectral gap of its Laplacian mutually control each other play an important role in graph theory and network analysis, in particular in the context of expander theory. The natural problem to extend such inequalities to simplicial complexes and their higher order Eckmann Laplacians has been open for a long time. Before proving any inequality, however, one needs to identify the right Cheeger-type constant for which such an inequality can hold. Here, we solve this problem. Our solution involves and combines constructions from simplicial topology, signed graphs, Gromov filling radii and an interpolation between the standard 2-Laplacians and the analytically more difficult 1-Laplacians, for which, however, the inequalities become equalities. It is then natural to develop a general theory for $p$-Laplacians on simplicial complexes and investigate the related Cheeger-type inequalities.

Cheeger inequalities on simplicial complexes

TL;DR

The paper addresses extending Cheeger inequalities to higher-dimensional simplicial complexes via the Eckmann up-Laplacian. By identifying a suitable Cheeger constant and employing a synthesis of simplicial topology, signed-graph theory, Gromov filling, and nonlinear p-Laplacian dualities, it proves a two-sided bound relating the first nonzero eigenvalue to a Cheeger constant for typical uniform triangulations of -manifolds. It also develops a complementary bound from the top eigenvalue using a signed-graph reduction and introduces a novel Cheeger constant based on generalized multisets to bound the ground-state eigenvalue of the up -Laplacian, with full p-Laplacian generality. These results provide a rigorous higher-dimensional Cheeger theory, linking geometry, topology, and nonlinear spectral analysis, and offering a framework for high-dimensional expander analysis and manifold triangulations. The approach combines graph-to-manifold approximation, nonlinear spectral duality, and a robust -Laplacian theory to yield universal-type inequalities that mirror the classical graph and manifold Cheeger inequalities in a higher-dimensional setting.

Abstract

Cheeger-type inequalities in which the decomposability of a graph and the spectral gap of its Laplacian mutually control each other play an important role in graph theory and network analysis, in particular in the context of expander theory. The natural problem to extend such inequalities to simplicial complexes and their higher order Eckmann Laplacians has been open for a long time. Before proving any inequality, however, one needs to identify the right Cheeger-type constant for which such an inequality can hold. Here, we solve this problem. Our solution involves and combines constructions from simplicial topology, signed graphs, Gromov filling radii and an interpolation between the standard 2-Laplacians and the analytically more difficult 1-Laplacians, for which, however, the inequalities become equalities. It is then natural to develop a general theory for -Laplacians on simplicial complexes and investigate the related Cheeger-type inequalities.
Paper Structure (12 sections, 16 theorems, 108 equations)

This paper contains 12 sections, 16 theorems, 108 equations.

Table of Contents

  1. Introduction and Background
  2. Simplicial complexes
  3. Cheeger inequalities
  4. Signed graphs
  5. A relation between simplicial complexes and signed graphs
  6. $p$-Laplacians
  7. Cheeger-type inequalities on $d$-faces of simplicial complexes
  8. Spectral gap from $d+2$
  9. Spectral gap from 0
  10. Background needed for the proof of Theorem \ref{['thm:Cheeger-manifold-complex']}
  11. Cheeger-type inequalities for $p$-Laplacians on simplicial complexes
  12. Discussions and Remarks

Key Result

Lemma 1.1

Let the linear operator $A:H\to H$ on a finite dimensional vector space be self-adjoint w.r.t. the scalar product $(.,.)$. Then its eigenvalues and eigenvectors are the critical values and the critical points of the Rayleigh quotient, defined for $f\neq 0$,

Theorems & Definitions (31)

  • Remark 1
  • Lemma 1.1
  • Lemma 1.2
  • Theorem 1.1: Atay20
  • Remark 2
  • Proposition 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Remark 3
  • Theorem 2.1
  • ...and 21 more