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

Abstract

A well known upper bound for the independence number $α(G)$ of a graph $G$, due to Cvetković, is that \begin{equation*} α(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 $α_q$(G), where $α_q(G) \ge α(G)$ and for some graphs $α_q(G) \gg α(G)$. 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 $k$.