  We derive eigenvalue bounds for the $t$-distance chromatic number of a graph,
which is a generalization of the classical chromatic number. We apply such
bounds to hypercube graphs, providing alternative spectral proofs for results
by Ngo, Du and Graham [Inf. Process. Lett., 2002], and improving their bound
for several instances. We also apply the eigenvalue bounds to Lee graphs,
extending results by Kim and Kim [Discrete Appl. Math., 2011]. Finally, we
provide a complete characterization for the existence of perfect Lee codes of
minimum distance $3$. In order to prove our results, we use a mix of spectral
and number theory tools. Our results, which provide the first application of
spectral methods to Lee codes, illustrate that such methods succeed to capture
the nature of the Lee metric.
