New Eigenvalue Bound for the Fractional Chromatic Number

TL;DR

This paper introduces a new eigenvalue-based lower bound for the fractional chromatic number of a graph, extending previous bounds for the chromatic number and employing association schemes in the proof.

Contribution

It presents a novel eigenvalue bound for the fractional chromatic number, strengthening prior results and generalizing to homomorphisms involving edge-transitive graphs.

Findings

Established a new eigenvalue bound for f(G)

Extended previous bounds from chromatic to fractional chromatic number

Utilized association schemes in the proof

Abstract

Given a graph $G$, we let $s^+(G)$ denote the sum of the squares of the positive eigenvalues of the adjacency matrix of $G$, and we similarly define $s^-(G)$. We prove that \[\chi_f(G)\ge 1+\max\left\{\frac{s^+(G)}{s^-(G)},\frac{s^-(G)}{s^+(G)}\right\}\] and thus strengthen a result of Ando and Lin, who showed the same lower bound for the chromatic number $\chi(G)$. We in fact show a stronger result wherein we give a bound using the eigenvalues of $G$ and $H$ whenever $G$ has a homomorphism to an edge-transitive graph $H$. Our proof utilizes ideas motivated by association schemes.