Some new trace formulas of tensors with applications in spectral hypergraph theory

We give some graph theoretical formulas for the trace (Formula presented.) of a tensor (Formula presented.) which do not involve the differential operators and auxiliary matrix. As applications of these trace formulas in the study of the spectra of uniform hypergraphs, we give a characterization (in terms of the traces of the adjacency tensors) of the (Formula presented.) -uniform hypergraphs whose spectra are (Formula presented.) -symmetric, thus give an answer to a question raised in Cooper and Dutle [Linear Algebra Appl. 2012;436:3268–3292]. We generalize the results in Cooper and Dutle [Linear Algebra Appl. 2012;436:3268–3292, Theorem 4.2] and Hu and Qi [Discrete Appl. Math. 2014;169:140–151, Proposition 3.1] about the (Formula presented.) -symmetry of the spectrum of a (Formula presented.) -uniform hypergraph, and answer a question in Hu and Qi [Discrete Appl. Math. 2014;169:140–151] about the relation between the Laplacian and signless Laplacian spectra of a (Formula presented.) -uniform hypergraph when (Formula presented.) is odd. We also give a simplified proof of an expression for (Formula presented.) and discuss the expression for (Formula presented.).