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

Abstract

Hoffman proved that a graph $G$ with eigenvalues $\mu_1 \geqslant \cdots \geqslant \mu_n$ and chromatic number $\chi(G)$ satisfies: \[\chi \geqslant 1 + \kappa\] where $\kappa$ is the smallest integer such that \[\mu_1 + \sum_{i=1}^{\kappa} \mu_{n+1-i} \leqslant 0.\] 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 $\chi_q(G) \leqslant \chi(G)$ 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 $\chi_q = \chi = p - 2$.

Spectral upper bound on the quantum $k$-independence number of a graph

“…The subject of quantum graph parameters has been extensively studied in the past decade, due to its connections to a number of subjects, including quantum information theory, operator theory, combinatorics and optimisation. The motivation for studying quantum graph parameters is described, for example, by Cameron et al [6] and in [18] and [9,10]. We add to this subject by defining an extension of the usual quantum independence number of a graph, called the quantum k-independence number, which is also motivated by its classical counterpart, the k-independence number of a graph.…”