Some properties of the Laplace and normalized Laplace spectra of uniform hypergraphs

In [8], Hu and Qi studied the normalized Laplace tensors and normalized Laplace spectra of k-uniform hypergraphs. They also mentioned the question about whether or not 2 is also an H-eigenvalue of the normalized Laplace tensor of a k-uniform hypergraph, when 2 is an eigenvalue of the normalized Laplace tensor (in this case, k is necessarily even). In this paper, we use an expression for the normalized Laplace tensor in terms of the tensor product, together with the diagonal similarity of tensors, the Perron–Frobenius Theorem for nonnegative tensors and nonnegative weakly irreducible tensors, and the concept and properties of odd-colorable hypergraphs introduced in [13], to give a complete answer to this question. We show that: (i). When k≡2(mod4), then the answer to this question is affirmative. (ii). When k≡0(mod4), then the answer to this question is negative, and in this case, we give an infinite family of counterexamples. We also study the signless normalized Laplace spectra and the signless normalized Laplace H-spectra of hypergraphs. We give structural characterizations of the hypergraphs having the same normalized Laplace spectrum and signless normalized Laplace spectrum, or having the same normalized Laplace H-spectrum and signless normalized Laplace H-spectrum, or both. Finally, we determine the first two k-uniform supertrees of order n with the largest Laplace spectral radii, and also determine the unique k-uniform hypertree of order n with the smallest Laplace spectral radii, in the case when k is even.