Spectra of Uniform Hypergraphs

TL;DR

This paper develops a spectral theory for uniform hypergraphs, extending concepts from spectral graph theory to hypermatrices, including defining eigenvalues and proving analogues of classical results.

Contribution

It introduces a framework for spectral analysis of hypergraphs using hyperdeterminants and eigenvalues of hypermatrices, advancing the understanding of hypergraph spectra.

Findings

Defined eigenvalues of hypermatrices via characteristic polynomial

Established analogues of spectral graph theory results for hypergraphs

Proposed open problems and future research directions

Abstract

We present a spectral theory of hypergraphs that closely parallels Spectral Graph Theory. A number of recent developments building upon classical work has led to a rich understanding of "hyperdeterminants" of hypermatrices, a.k.a. multidimensional arrays. Hyperdeterminants share many properties with determinants, but the context of multilinear algebra is substantially more complicated than the linear algebra required to address Spectral Graph Theory (i.e., ordinary matrices). Nonetheless, it is possible to define eigenvalues of a hypermatrix via its characteristic polynomial as well as variationally. We apply this notion to the "adjacency hypermatrix" of a uniform hypergraph, and prove a number of natural analogues of basic results in Spectral Graph Theory. Open problems abound, and we present a number of directions for further study.