New Eigenvalue Bound for the Fractional Chromatic Number

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 $${\chi }_f(G)\ge 1+\max \{\frac{s^{+}(G)}{s^{-}(G)},\frac{s^{-}(G)}{s^{+}(G)}\}$$ 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.