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

TL;DR

This paper extends spectral lower bounds from classical to quantum chromatic numbers, showing that quantum chromatic number can be significantly smaller for certain graphs, with implications for graph theory and quantum computing.

Contribution

It generalizes Hoffman’s spectral bound to quantum chromatic numbers and explores its implications for various classes of graphs.

Findings

Quantum chromatic number can be strictly less than classical chromatic number.

Spectral bounds apply to quantum chromatic number, providing new lower bounds.

Kneser graphs have equal quantum and classical chromatic numbers, matching classical bounds.

Abstract

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 $\chi_q = \chi = p - 2$.