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

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 $(n^+, n^0, n^-)$istheinertiaof$G$. We prove that this bound is also an upper bound for the quantum independence number ${\alpha }_q$(G), where $\alpha_q(G) \ge \alpha(G)$\mathrm{.}Weidentifynumerousgraphsforwhich$\alpha(G) = \alpha_q(G)$ 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.