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Normalized Laplacian eigenvalues of hypergraphs

Leyou Xu, Bo Zhou

Abstract

In this paper, we give tight bounds for the normalized Laplacian eigenvalues of hypergraphs that are not necessarily uniform, and provide an edge version interlacing theorem, a Cheeger inequality, and a discrepancy inequality that are related to the normalized Laplacian eigenvalues for uniform hypergraphs.

Normalized Laplacian eigenvalues of hypergraphs

Abstract

In this paper, we give tight bounds for the normalized Laplacian eigenvalues of hypergraphs that are not necessarily uniform, and provide an edge version interlacing theorem, a Cheeger inequality, and a discrepancy inequality that are related to the normalized Laplacian eigenvalues for uniform hypergraphs.
Paper Structure (6 sections, 15 theorems, 90 equations)

This paper contains 6 sections, 15 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Bounds for the normalized Laplacian eigenvalues
  4. Interlacing theorem
  5. Cheeger inequality
  6. Discrepancy inequality

Key Result

Theorem 2.1

If $H$ is a hypergraph on $n$ vertices, then $\mathcal{L}(H)$ is semi-definite, $\lambda_n(H)=0$ with a corresponding eigenvector $D(H)^{1/2}\mathbf{1}_n$, and the multiplicity of $\lambda_n(H)=0$ is exactly the number of components of $H$.

Theorems & Definitions (28)

  • Theorem 2.1
  • proof
  • Lemma 2.1
  • proof
  • Theorem 3.1
  • proof
  • Corollary 3.1
  • proof
  • Corollary 3.2
  • Theorem 3.2
  • ...and 18 more