New Eigenvalue Bound for the Fractional Chromatic Number - Sam Spiro

Title: 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 \[\chi_f(G)\ge 1+\max\left\{\frac{s^+(G)}{s^-(G)},\frac{s^-(G)}{s^+(G)}\right\}\] 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 (though requires no knowledge of them).  This is joint work with Krystal Guo.