Spectral Lower Bounds for the Quantum Chromatic Number of a Graph -- Part II

Hoffman proved that a graph $G$ with eigenvalues $\mu_1 \ge \ldots \ge \mu_n$ and chromatic number $\chi(G)$ satisfies: $$\chi \ge 1 + \kappa$$ where $\kappa$ is the smallest integer such that $$\mu_1 + \sum_{i=1}^{\kappa} \mu_{n+1-i} \le 0.$$ We strengthen this well known result by proving that $\chi(G)$ can be replaced by the quantum chromatic number, $\chi_q(G)$, where for all graphs $\chi_q(G) \le \chi(G)$ and for some graphs $\chi_q(G)$ is significantly smaller than $\chi(G)$. We also prove a similar result, and investigate implications of these inequalities for the quantum chromatic number of various classes of graphs, which improves many known results. For example, we demonstrate that the Kneser graph $KG_{p,2}$ has $\chi_q = \chi = p - 2$.