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

χf (G) ≥ 1 + max { s+G/s-G, s-G/s+G}

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 has a homomorphism to an edge-transitive graph
. Our proof utilizes ideas motivated by association schemes.