Spectra of power hypergraphs and signed graphs via parity-closed walks

The k-power hypergraph $G^{\left(k\right)}$ is the k-uniform hypergraph that is obtained by adding $k-2$ new vertices to each edge of a graph G, for $k\ge 3$. A parity-closed walk in G is a closed walk that uses each edge an even number of times. In an earlier paper, we determined the eigenvalues of the adjacency tensor of $G^{\left(k\right)}$ using the eigenvalues of signed subgraphs of G. Here, we express the entire spectrum (that is, we determine all multiplicities and the characteristic polynomial) of $G^{\left(k\right)}$ in terms of parity-closed walks of G. Moreover, we give an explicit expression for the multiplicity of the spectral radius of $G^{\left(k\right)}$. As a side result, we show that the number of parity-closed walks of given length is the corresponding spectral moment averaged over all signed graphs with underlying graph G. By extrapolating the characteristic polynomial of $G^{\left(k\right)}$ to $k=2$, we introduce a pseudo-characteristic function which is shown to be the geometric mean of the characteristic polynomials of all signed graphs on G. This supplements a result by Godsil and Gutman that the arithmetic mean of the characteristic polynomials of all signed graphs on G equals the matching polynomial of G.