Spectral upper bound on the quantum $k$-independence number of a graph

Abstract

A well known upper bound for the independence number $\alpha(G)$ of a graph $G$, due to Cvetkovi ́c, is that
\begin{equation*}
\alpha(G) \le n^0 + \min\{n^+ , n^-\}
\end{equation*}
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 $\alpha_q(G) \ge \alpha(G)$ and for some graphs $\alpha_q(G) \gg \alpha(G)$. We identify numerous graphs for which $\alpha(G) = \alpha_q(G)$, thus increasing the number of graphs for which $\alpha_q$ is known. We also demonstrate that there are graphs for which the above bound is not exact with any Hermitian weight matrix, for $\alpha(G)$ and $\alpha_q(G)$. Finally, we show this result in the more general context of spectral bounds for the quantum $k$-independence number, where the $k$-independence number is the maximum size of a set of vertices at pairwise distance greater than $k$.