Work

Eigenvalue bounds for the distance-t chromatic number of a graph and their application to Lee codes

Year: 2025

Type: article

Abstract: We derive eigenvalue bounds for the distance-[Formula: see text] 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 [New bounds on a hypercube coloring problem, Inform. Process. Lett. 84(5) (2002) 265–269], and improving their bound for several instances. We also apply the eigenvalue bounds to Lee graphs, extending results by Kim and Kim [The [Formula: see text]-distance coloring of the Cartesian product of cycles using optimal Lee codes, Discr. Appl. Math. 159(18) (2011) 2222–2228]. Finally, we provide a complete characterization for the existence of perfect Lee codes of minimum distance [Formula: see text]. 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 spectral graph theory succeeds to capture the nature of the Lee metric.

Source: Journal of Algebra and Its Applications

DOI: 10.1142/s0219498825410245

Authors: Aida Abiad, Alessandro Neri, Luuk Reijnders

Institutions: Vrije Universiteit Brussel, Eindhoven University of Technology, Ghent University, University of Naples Federico II

Language: English

Topic: Graph theory and applications

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: Fonds Wetenschappelijk Onderzoek, Nederlandse Organisatie voor Wetenschappelijk Onderzoek, Vlaamse regering

Awards: Finding network structure beyond the spectrum: new frontiers and new methods, New Geometric and Algebraic Foundations of Coding Theory