Orthonormal representation of a graph

Assign a real unit vector $u_v$ to each graph vertex, with $u_v\perp u_w$ whenever distinct vertices are nonadjacent. A handle is a further unit vector $h$. In the handle formulation of the Lovász number, one minimizes ${\max }_v|h\cdot u_v{|}^{-2}$ over representations and handles. Orthonormal here does not mean every pair of assigned vectors is orthogonal.

Tensor products of the assigned vectors represent strong graph powers, and tensor products of the handle have squared inner products equal to products of the original ones. The vectors assigned to an independent set are orthonormal. Bessel inequality then bounds its size by the largest inverse squared handle inner product. Taking roots and minimizing proves $c(G)\le \vartheta (G)$.