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

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.

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