The rank of connection matrices and the dimension of graph algebras

Connection matrices were introduced in [M. Freedman, L. Lovلsz, A. Schrijver, Reflection positivity, rank connectivity, and homomorphism of graphs (MSR Tech Report # MSR-TR-2004-41) ftp://ftp.research.microsoft.com/pub/tr/TR-2004-41.pdf], where they were used to characterize graph homomorphism functions. The goal of this note is to determine the exact rank of these matrices. The result can be rephrased in terms of the dimension of graph algebras, also introduced in the same paper. Yet another version proves that if two k -tuples of nodes behave in the same way from the point of view of graph homomorphisms, then they are equivalent under the automorphism group.