New Eigenvalue Bound for the Fractional Chromatic Number
Krystal Guo, Sam Spiro
TL;DR
This work proves a new spectral lower bound for the fractional chromatic number: $\chi_f(G)\ge 1+\max\{\frac{s^+(G)}{s^-(G)},\frac{s^-(G)}{s^+(G)}\}$, where $s^+(G)$ and $s^-(G)$ sum the squares of positive and negative adjacency eigenvalues, respectively. It extends the classical Ando–Lin bound from the ordinary chromatic number to $\chi_f(G)$ and strengthens it via a general framework: if $G$ maps to an edge-transitive graph $H$, then $\frac{\lambda_{\max}(H)}{|\lambda_{\min}(H)|}$ upper-bounds the relevant eigenvalue-ratio of $G$. The proofs combine a partition-into-fibres method with a lemma grounded in the theory of association schemes, enabling a decomposition $A=X-Y$ into PSD parts and translating spectral information through the $H$-partition. The results highlight tightness in several graphs, compare with Hoffman's and clique bounds, and point to open questions about extending to the vector chromatic number $\chi_c(G)$ and broader families of target graphs $H$.
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 \[χ_f(G)\ge 1+\max\left\{\frac{s^+(G)}{s^-(G)},\frac{s^-(G)}{s^+(G)}\right\}\] and thus strengthen a result of Ando and Lin, who showed the same lower bound for the chromatic number $χ(G)$. 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.