# An inertial upper bound for the quantum independence number of a graph. (arXiv:1808.10820v4 [math.CO] UPDATED)

$G$, 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.