The spectral property of hypergraph coverings

TL;DR

This paper explores how the spectral properties of hypergraph coverings relate to the original hypergraph, providing characterizations, divisibility relations, and explicit formulas, especially for 2-fold coverings, using algebraic and combinatorial methods.

Contribution

It characterizes the connectedness of hypergraph coverings and establishes spectral relations between a hypergraph and its coverings, including explicit formulas for 2-fold cases.

Findings

Connectedness characterized by incidence graph and permutation assignment

Spectral indices of coverings are divisible by those of the original hypergraph

Spectrum of 2-fold coverings contains the original spectrum and that of a signed hypergraph

Abstract

Let $H$ be a connected $m$-uniform hypergraph, and let $\mathcal{A}(H)$ be the adjacency tensor of $H$ whose spectrum is simply called the spectrum of $H$. Let $s(H)$ denote the number of eigenvectors of $\mathcal{A}(H)$ associated with the spectral radius, and $c(H)$ denote the number of eigenvalues of $\mathcal{A}(H)$ with modulus equal to the spectral radius, which are respectively called the stabilizing index and cyclic index of $H$. Let $\bar{H}$ be a $k$-fold covering of $H$ which can be obtained from some permutation assignment in the symmetric group $\mathbf{S}_k$ on $H$. In this paper, we first characterize the connectedness of $\bar{H}$ by its incidence graph and the permutation assignment, and then investigate the relationship between the spectral property of $H$ and that of $\bar{H}$. By applying module theory and group representation, if $\bar{H}$ is connected, we prove that $s(H) \mid s(\bar{H})$ and $c(H) \mid c(\bar{H})$. In particular, when $\bar{H}$ is a $2$-fold covering of $H$, if $m$ is even, we show that regardless of multiplicities, the spectrum of $\bar{H}$ contains the spectrum of $H$ and the spectrum of a signed hypergraph with $H$ as underlying hypergraph; if $m$ is odd, we give an explicit formula for $s(\bar{H})$. We also find some differences on the spectral property between hypergraph coverings and graph coverings by examples.