Authors’ abstract: We give some graph theoretical formulas for the trace \(\mathrm{Tr}_k(\mathbb{T})\) of a tensor \(\mathbb{T}\) 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, they give a characterization (in terms of the traces of the adjacency tensors) of the \(k\)-uniform hypergraphs whose spectra are \(k\)-symmetric, thus give an answer to a question raised by J. Cooper and A. Dutle [Linear Algebra Appl. 436, No. 9, 3268–3292 (2012; Zbl 1238.05183)]. We generalize the results by Cooper and Dutle [loc. cit., Theorem 4.2] and by S. Hu and L. Qi [Discrete Appl. Math. 169, 140–151 (2014; Zbl 1288.05157), Proposition 3.1] about the \(k\)-symmetry of the spectrum of a \(k\)-uniform hypergraph, and answer a question of Hu and Qi [loc. cit.] about the relation between the Laplacian and signless Laplacian spectra of a \(k\)-uniform hypergraph when \(k\) is odd. They also give a simplified proof of an expression for \(\mathrm{Tr}_2(\mathbb{T})\) and discuss the expression for \(\mathrm{Tr}_3(\mathbb{T})\).