On orthogonal representations of graphs

Let G be a simple graph with vertices 1, 2, …, n and F be a field. We consider representation of G by vectors x1, x2, …, xn∈Fd such that for i≠j the product xixj=xi1xj1+xi2xj2+⋯+xidxjd is equal to zero if and only if vertices i and j are not adjacent in G. The least dimension d necessary for such representations is studied as a function of G.