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.

翻译：我们推导了图的$t$-距离色数的特征值界，这是经典色数概念的一种推广。我们将这些界应用于超立方体图，为Ngo、Du和Graham [Inf. Process. Lett., 2002] 的结果提供了替代的谱方法证明，并在若干实例中改进了他们的界。我们还将特征值界应用于Lee图，扩展了Kim和Kim [Discrete Appl. Math., 2011] 的研究结果。最后，我们完整刻画了最小距离为$3$的完美Lee码的存在条件。为证明我们的结论，我们综合运用了谱方法与数论工具。这些结果首次将谱方法应用于Lee码，并表明此类方法能够有效刻画Lee度量的本质特性。