New Eigenvalue Bound for the Fractional Chromatic Number

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 $\chi (G)$. 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.

11 pages

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