Spectral Lower Bounds for the Quantum Chromatic Number of a Graph -- Part II
Hoffman proved that a graph $G$ with eigenvalues $μ_1 \ge \ldots \ge μ_n$ and chromatic number $χ(G)$ satisfies: \[ χ\ge 1 + κ\] where $κ$ is the smallest integer such that \[ μ_1 + \sum_{i=1}^κ μ_{n+1-i} \le 0. \] We strengthen this well known result by proving that $χ(G)$ can be replaced by the quantum chromatic number, $χ_q(G)$, where for all graphs $χ_q(G) \le χ(G)$ and for some graphs $χ_q(G)$ is significantly smaller than $χ(G)$. 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 $χ_q = χ= p - 2$.
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