The spectral property of hypergraph coverings.

Let H be a connected m -uniform hypergraph, and let 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 A (H) associated with the spectral radius, and c (H) denote the number of eigenvalues of A (H) with modulus equal to the spectral radius, which are respectively called the stabilizing index and cyclic index of H. Let H ¯ be a k -fold covering of H which can be obtained from some permutation assignment in the symmetric group S k on H. In this paper, we first characterize the connectedness of H ¯ by its incidence graph and the permutation assignment, and then investigate the relationship between the spectral property of H and that of H ¯. By applying module theory and group representation, if H ¯ is connected, we prove that s (H) | s (H ¯) and c (H) | c (H ¯). In particular, when H ¯ is a 2-fold covering of H , if m is even, we show that regardless of multiplicities, the spectrum of 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 (H ¯). We also find some differences on the spectral property between hypergraph coverings and graph coverings by examples. [ABSTRACT FROM AUTHOR]