Adjacency Spectra of Random and Uniform Hypergraphs

TL;DR

This paper investigates the asymptotic behavior of adjacency eigenvalues in random and complete uniform hypergraphs, connecting their spectra to all-ones hypermatrices and establishing bounds and approximations.

Contribution

It advances understanding of hypergraph spectra by relating them to all-ones hypermatrices and providing bounds on spectral radius for specific hypergraph ensembles.

Findings

Bound on spectral radius of symmetric Bernoulli hyperensemble

Spectrum of complete 2- and 3-uniform hypergraphs approximates scaled all-ones hypermatrix

Progress towards conjecture linking hypergraph spectra to all-ones hypermatrices

Abstract

We present progress on the problem of asymptotically describing the adjacency eigenvalues of random and complete uniform hypergraphs. There is a natural conjecture arising from analogy with random matrix theory that connects these spectra to that of the all-ones hypermatrix. Several of the ingredients along a possible path to this conjecture are established, and may be of independent interest in spectral hypergraph/hypermatrix theory. In particular, we provide a bound on the spectral radius of the symmetric Bernoulli hyperensemble, and show that the spectrum of the complete \(k\)-uniform hypergraph for \(k=2,3\) is close to that of an appropriately scaled all-ones hypermatrix.