An inertial upper bound for the quantum independence number of a graph

TL;DR

This paper extends a classical inertia-based upper bound for the independence number of a graph to the quantum independence number, showing it holds in the quantum setting and identifying cases of equality and discrepancy.

Contribution

The authors prove that the inertia-based upper bound applies to the quantum independence number and analyze cases where it is tight or not, advancing understanding of quantum graph parameters.

Findings

The inertia bound is valid for the quantum independence number.

There are graphs where the bound is tight for both classical and quantum independence numbers.

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 $G$, is that \[ \alpha(G) \le n^0 + \min\{n^+ , n^-\}, \] 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 $\alpha_q(G) \ge \alpha(G)$. We identify numerous graphs for which $\alpha(G) = \alpha_q(G)$ and demonstrate that there are graphs for which the above bound is not exact with any Hermitian weight matrix, for $\alpha(G)$ and $\alpha_q(G)$. This result complements results by the authors that many spectral lower bounds for the chromatic number are also lower bounds for the quantum chromatic number.