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

Abstract

The orthogonal rank of a graph $G=(V,E)$ is the smallest dimension $ξ$ such that there exist non-zero column vectors $x_v\in\mathbb{C}^ξ$ 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, $χ$, are also lower bounds for $ξ$. This result complements a previous result by the authors, in which they showed that spectral lower bounds for $χ$ are also lower bounds for the quantum chromatic number $χ_q$. 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 $ξ_f$, and conjecture that a stronger inertial lower bound for $ξ$ is also a lower bound for $ξ_f$.

Improved proof of lower bound on orthogonal rank (Theorem 4); authors appreciate comments