Spectral lower bounds for the orthogonal and projective ranks of a graph

TL;DR

This paper establishes spectral lower bounds for the orthogonal and projective ranks of a graph, linking them to spectral bounds for chromatic number and introducing inertial bounds for the projective rank.

Contribution

It proves that spectral bounds for chromatic number also serve as bounds for the orthogonal rank and introduces inertial lower bounds for the projective rank.

Findings

Spectral lower bounds for chromatic number are also lower bounds for orthogonal rank.

An inertial lower bound for the projective rank is established.

Conjecture that inertial bounds for orthogonal rank extend to the projective rank.

Abstract

The orthogonal rank of a graph $G=(V,E)$ is the smallest dimension $\xi$ such that there exist non-zero column vectors $x_v\in\mathbb{C}^\xi$ for $v\in V$ satisfying the orthogonality condition $x_v^\dagger x_w=0$ for all $vw\in E$. We prove that many spectral lower bounds for the chromatic number, $\chi$, are also lower bounds for $\xi$. This result complements a previous result by the authors, in which they showed that spectral lower bounds for $\chi$ are also lower bounds for the quantum chromatic number $\chi_q$. It is known that the quantum chromatic number and the orthogonal rank are incomparable. We conclude by proving an inertial lower bound for the projective rank $\xi_f$, and conjecture that a stronger inertial lower bound for $\xi$ is also a lower bound for $\xi_f$.