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

A well known upper bound for the independence number α(G)α(G) of a graph GG, due to Cvetkovi ́c, is that α(G)≤n0+min{n+,n−}α(G)≤n0+min{n+,n−} where (n+,n0,n−)(n+,n0,n−) is the inertia of GG. We prove that this bound is also an upper bound for the quantum independence number αqαq(G), where αq(G)≥α(G)αq(G)≥α(G) and for some graphs αq(G)≫α(G)αq(G)≫α(G). We identify numerous graphs for which α(G)=αq(G)α(G)=αq(G), thus increasing the number of graphs for which αqαq 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)α(G) and αq(G)αq(G). Finally, we show this result in the more general context of spectral bounds for the quantum kk-independence number, where the kk-independence number is the maximum size of a set of vertices at pairwise distance greater than kk