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

TL;DR

This paper extends classical spectral bounds to the quantum setting, providing new upper bounds for the quantum independence number of graphs and exploring their tightness and limitations.

Contribution

It proves a spectral upper bound for the quantum independence number, generalizes to quantum k-independence, and identifies graphs where the bound is tight or not.

Findings

The spectral bound applies to quantum independence number $ abla_q(G)$.

The bound is tight for many graphs where $ abla_q(G) = abla(G)$.

Some graphs do not achieve the bound with any Hermitian weight matrix.

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 $k$.