On the spectrum of hypergraphs

TL;DR

This paper explores the spectral properties of hypergraphs through various matrices, linking spectral characteristics to structural properties, connectivity, and curvature, and introduces methods for analyzing random walks and Ricci curvature on hypergraphs.

Contribution

It introduces a comprehensive spectral framework for hypergraphs, including bounds on connectivity, diameter, chromatic number, Cheeger constant, and Ricci curvature, extending spectral graph theory to hypergraphs.

Findings

Spectral radii are bounded by hypergraph degrees.

Hypergraph diameter is bounded by eigenvalues of connectivity matrices.

Cheeger constant can be bounded by Laplacian eigenvalues.

Abstract

Here we study the spectral properties of an underlying weighted graph of a non-uniform hypergraph by introducing different connectivity matrices, such as adjacency, Laplacian and normalized Laplacian matrices. We show that different structural properties of a hypergrpah, can be well studied using spectral properties of these matrices. Connectivity of a hypergraph is also investigated by the eigenvalues of these operators. Spectral radii of the same are bounded by the degrees of a hypergraph. The diameter of a hypergraph is also bounded by the eigenvalues of its connectivity matrices. We characterize different properties of a regular hypergraph characterized by the spectrum. Strong (vertex) chromatic number of a hypergraph is bounded by the eigenvalues. Cheeger constant on a hypergraph is defined and we show that it can be bounded by the smallest nontrivial eigenvalues of Laplacian matrix and normalized Laplacian matrix, respectively, of a connected hypergraph. We also show an approach to study random walk on a (non-uniform) hypergraph that can be performed by analyzing the spectrum of transition probability operator which is defined on that hypergraph. Ricci curvature on hypergraphs is introduced in two different ways. We show that if the Laplace operator, $\Delta$, on a hypergraph satisfies a curvature-dimension type inequality $CD (\mathbf{m}, \mathbf{K})$ with $\mathbf{m}>1 $ and $\mathbf{K}>0 $ then any non-zero eigenvalue of $- \Delta$ can be bounded below by $ \frac{ \mathbf{m} \mathbf{K}}{ \mathbf{m} -1 } $. Eigenvalues of a normalized Laplacian operator defined on a connected hypergraph can be bounded by the Ollivier's Ricci curvature of the hypergraph.