SPECTRAL UPPER BOUND ON THE QUANTUM K-INDEPENDENCE NUMBER OF A GRAPH

well-known upper bound for the independence number alpha(G) of a graph G, due to Cvetkovic, is that alpha(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 alpha(q)(G), where alpha(q)(G) >= alpha(G) and for some graphs alpha(q)(G) >> alpha(G). We identify numerous graphs for which alpha(G) = alpha(q)(G), thus increasing the number of graphs for which a 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.