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 $\xi_f$.
Highlights
We conclude by proving an inertial lower bound for the projective rank ξf, and conjecture that a stronger inertial lower bound for ξ is a lower bound for ξf
For any graph G, let V denote the set of vertices where |V | = n, E denote the set of edges where |E| = m, A denote the adjacency matrix, χ(G) denote the chromatic number, ω(G) denote the clique number, α(G) denote the independence number, and G denote the complement of G
Let D be the diagonal matrix of vertex degrees, and let L = D − A denote the Laplacian of G and Q = D + A denote the signless Laplacian of G
Summary
Let χv(G) and χsv(G) denote the vector and strict vector chromatic numbers as defined by Karger et al [14]. They proved that χsv(G) = θ(G), where θ is the Lovasz theta the electronic journal of combinatorics 26(3) (2019), #P3.45 function [16]. The orthogonal rank of G is the smallest positive integer ξ(G) such that there exists an orthogonal representation, that is a collection of non-zero column vectors xv ∈ Cξ(G) for v ∈ V satisfying the orthogonality condition x†vxw = 0. The least integer c for which there exists a vectorial c-coloring will be denoted χvect(G) and called the vectorial chromatic number of G.
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