On orthogonal representations of graphs

Abstract

Let G be a simple graph with vertices 1, 2, …, n and F be a field. We consider representation of G by vectors x 1, x 2, …, x n∈F d such that for i≠j the product x ix j=x i 1x j 1+x i 2x j 2+⋯+x i dx j d 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.