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For a k-uniform hypergraph H, we obtain some trace formulas for the Laplacian tensor of H, which imply that n i=1 d s i (s = 1, . . . , k) is determined by the Laplacian spectrum of H, where d 1 , . . . , d n is the degree sequence of H. Using trace formulas for the Laplacian tensor, we obtain expressions for some coefficients of the Laplacian polynomial of a regular hypergraph. We give some spectral characterizations of odd-bipartite hypergraphs, and give a partial answer to a question posed by Shao et al [17]. We also give some spectral properties of power hypergraphs, and show that a conjecture posed by Hu et al [7] holds under certain conditons.

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By using digraphs of tensors, we give Brualdi-type eigenvalue inclusion sets of tensors. We also give some applications of our result to nonsingularity and positive definiteness of tensors.