Hypergraph coverings and Ramanujan Hypergraphs

TL;DR

This paper explores the spectral properties of Ramanujan hypergraphs using hypergraph coverings, establishing bounds and existence results for Ramanujan hypergraphs through advanced polynomial techniques.

Contribution

It generalizes spectral bounds from graphs to hypergraphs and proves the existence of infinite families of Ramanujan hypergraphs using hypergraph coverings and polynomial interlacing methods.

Findings

Spectrum of hypergraph coverings includes original spectrum and additional spectra

Lower bounds for second largest eigenvalue in hypergraphs are established

Existence of infinite Ramanujan hypergraph families is demonstrated

Abstract

In this paper we investigate Ramanujan hypergraphs by using hypergraph coverings. We first show that the spectrum of a $k$-fold covering $\bar{H}$ of a connected hypergraph $H$ contains the spectrum of $H$, and that it is the union of the spectrum of $H$ and the spectrum of an incidence-signed hypergraph with $H$ as underlying hypergraph if $k=2$, which generalizes Bilu-Linial result on graph coverings. We give a lower bound for the second largest eigenvalue of a $d$-regular hypergraph by universal cover, which generalizes Alon-Boppana bound on $d$-regular graphs and Feng-Li bound on $(d,r)$-regular hypergraphs. By using interlacing family of polynomials, we prove that every $(d,r)$-regular hypergraph has a right-sided Ramanujan $2$-covering, and has a left-sided Ramanujan $2$-covering if the roots of the matching polynomial of its incident graph satisfy some condition. By Ramanujan $2$-coverings, we prove the existence of some families of infinite many left-sided or right-sided $(d,r)$-regular Ramanujan hypergraphs under certain conditions on $d$ and $r$.