An inertial upper bound for the quantum independence number of a graph

Abstract

A well known upper bound for the independence number $\alpha(G)$ of a graph $G$, is that \[ \alpha(G) \le n^0 + \min\{n^+ , n^-\}, \] where $(n^+, n^0, n^-)$ is the inertia of $G$. We prove that this bound is also an upper bound for the quantum independence number $\alpha_q$(G), where $\alpha_q(G) \ge \alpha(G)$. We identify numerous graphs for which $\alpha(G) = \alpha_q(G)$ and demonstrate that there are graphs for which the above bound is not exact with any Hermitian weight matrix, for $\alpha(G)$ and $\alpha_q(G)$. This result complements results by the authors that many spectral lower bounds for the chromatic number are also lower bounds for the quantum chromatic number.