  We give some graph theoretical formulas for the trace $Tr_k(\mathbb {T})$ 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, we 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 in [3]. We
generalize the results in [3, Theorem 4.2] and [5, Proposition 3.1] about the
$k$-symmetry of the spectrum of a $k$-uniform hypergraph, and answer a question
in [5] about the relation between the Laplacian and signless Laplacian spectra
of a $k$-uniform hypergraph when $k$ is odd. We also give a simplified proof of
an expression for $Tr_2(\mathbb {T})$ and discuss the expression for
$Tr_3(\mathbb {T})$.
