TY - JOUR
T1 - Eigenvalue bounds for the distance-t chromatic number of a graph and their application to perfect Lee codes
AU - Abiad Monge, Aida
AU - Neri, Alessandro
AU - Reijnders, Luuk E.R.M.
PY - 2024
Y1 - 2024
N2 - 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.
AB - 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.
M3 - Article
SN - 0219-4988
VL - XX
JO - Journal of Algebra and its Applications
JF - Journal of Algebra and its Applications
IS - X
ER -