Spectral Lower Bounds for the Orthogonal and Projective Ranks of a Graph

The orthogonal rank of a graph G = (V, E) is the smallest dimension xi such that there exist non-zero column vectors x(v) is an element of C-xi for v is an element of V satisfying the orthogonality condition x(v)(dagger)x(w) = 0 for all vw is an element of 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).