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

TL;DR

This paper extends graph coloring and independence concepts to operators on L^2 spaces, providing spectral bounds that connect harmonic analysis and convex optimization for finite and infinite graphs.

Contribution

It introduces a spectral framework for bounding independence ratio and chromatic number of operators, generalizing finite graph bounds to infinite and geometric graphs.

Findings

Bounds for independence ratio and chromatic number in terms of the operator's numerical range

Application of the theory to infinite geometric graphs on Euclidean space and the sphere

A unified analytical approach for packing and coloring problems using harmonic analysis

Abstract

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.