Spectral Lower Bounds For The Quantum Chromatic Number Of A Graph - Part Ii

Hoffman proved that a graph G with eigenvalues mu(1) >= ... >= mu(n) and chromatic number chi(G) satisfies:chi >= 1 + kappawhere kappa is the smallest integer such thatmu(1) + Sigma(kappa)(i=1) mu(n)+1-i <= 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) <= 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.