Spectral bounds for the independence ratio and the chromatic number of an operator

Christine Bachoc (1) , Evan Decorte , Fernando Mario de Oliveira Filho , Frank Vallentin (2)

1 IMB - Institut de Mathématiques de Bordeaux
2 TWA - Delft Institute of Applied Mathematics

We define the independence ratio and the chromatic number for bounded, self-adjoint operators on an L^2-space by extending the definitions for the adjacency matrix of finite graphs. In analogy to the Hoffman bounds for finite graphs, we give bounds for these parameters in terms of the numerical range of the operator. This provides a theoretical framework in which many packing and coloring problems for finite and infinite graphs can be conveniently studied with the help of harmonic analysis and convex optimization. The theory is applied to infinite geometric graphs on Euclidean space and on the unit sphere.

Domaines: Analyse fonctionnelle [math.FA]; Combinatoire [math.CO]; Optimisation et contrôle [math.OC]

Christine Bachoc, Evan Decorte, Fernando Mario de Oliveira Filho, Frank Vallentin. Spectral bounds for the independence ratio and the chromatic number of an operator. 2013. ⟨hal-00911679⟩