The spectral property of hypergraph coverings

Let $H$ be connected $m$-uniform hypergraph and $mathcal{A}(H)$ be the adjacency tensor of $H$. The stabilizing index of $H$, denoted by $s(H)$, is exactly the number of eigenvectors of $mathcal{A}(H)$ associated with the spectral radius, and the cyclic index of $H$, denoted by $c(H)$, is the number of eigenvalues of $mathcal{A}(H)$ with modulus equal to the spectral radius. Let $bar{H}$ be a $k$-fold covering of $H$. Then $bar{H}$ is isomorphic to a hypergraph $H_B^phi$ derived from the incidence graph $B_H$ of $H$ together with a permutation voltage assignment $phi$ in the symmetric group $mathbb{S}_k$. In this paper, we first characterize the connectedness of $bar{H}$ by using $H_B^phi$ for subsequent discussion. By applying the theory of module and group representation, we prove that if $bar{H}$ is connected, then $s(H) mid s(bar{H})$ and $c(H) mid c(bar{H})$. In the situation that $bar{H}$ is a $2$-fold covering of $H$, if $m$ is even, we show that regardless of multiplicities, the spectrum of $mathcal{A}(bar{H})$ contains the spectrum of $mathcal{A}(H)$ and the spectrum of a signed hypergraph constructed from $H$ and the covering projection; if $m$ is odd, we give an explicit formula for $s(bar{H})$.