Unified spectral bounds on the chromatic number

TL;DR

This paper extends spectral bounds on the chromatic number of graphs by incorporating all eigenvalues of various matrices, unifying existing bounds and demonstrating improved results in certain cases.

Contribution

It generalizes spectral bounds on the chromatic number to include all eigenvalues of adjacency, Laplacian, and signless Laplacian matrices, unifying multiple known bounds.

Findings

New bounds outperform existing bounds in specific examples

Unified spectral bounds apply to multiple matrix types

Enhanced understanding of spectral properties related to graph coloring

Abstract

One of the best known results in spectral graph theory is the following lower bound on the chromatic number due to Alan Hoffman, where mu_1 and mu_n are respectively the maximum and minimum eigenvalues of the adjacency matrix: chi >= 1 + mu_1 / (- mu_n). We recently generalised this bound to include all eigenvalues of the adjacency matrix. In this paper, we further generalize these results to include all eigenvalues of the adjacency, Laplacian and signless Laplacian matrices. The various known bounds are also unified by considering the normalized adjacency matrix, and examples are cited for which the new bounds outperform known bounds.