An eigenvalue bound for the fractional chromatic number

TL;DR

This paper establishes a spectral bound for the fractional chromatic number of graphs, connecting eigenvalue sums with the Lovász theta number and its generalizations, providing insights into graph coloring bounds.

Contribution

It demonstrates that Hoffman's eigenvalue sum bound is comparable to the Lovász theta number and relates it to the fractional chromatic number through a novel connection.

Findings

Eigenvalue sum bound is at least as good as Lovász theta number

Bound is no better than the ceiling of the fractional chromatic number

Connects eigenvalue bounds with a generalization of Lovász theta number

Abstract

We show that Hoffman's sum of eigenvalues bound for the chromatic number is at least as good as the Lov\'asz theta number, but no better than the ceiling of the fractional chromatic number. In order to do so, we display an interesting connection between this sum of eigenvalues bound and a generalization of the Lov\'asz theta number introduced by Manber and Narasimhan in 1988.