  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.
