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

A well-known upper bound for the independence number α(G) of a graph G, due to Cvetković, is that α(G) ≤ 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 α _q(G), where α _q(G) ≥ α(G) and for some graphs α _q(G) ≫ α(G). We identify numerous graphs for which α(G) = α _q(G), thus increasing the number of graphs for which α _q is known. We also demonstrate that there are graphs for which the above bound is not exact with any Hermitian weight matrix, for α(G) and α _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.