We characterize the rank of edge connection matrices of partition functions of real vertex models, as the dimension of the homogeneous components of the algebra of @math -invariant tensors. Here @math is the sub- group of the real orthogonal group that stabilizes the vertex model. This answers a question of Bal 'azs Szegedy from 2007.
Given a symmetric nxn matrix A, we define, for any graph G,f"A(G):=@?@f:VG-> 1,...,n @?uv@?EGa"@f"("u")","@f"("v"). We characterize for which graph parameters f there is a complex matrix A with f=f"A, and similarly for real A. We show that f"A uniquely determines A, up to permuting rows and (simultaneously) columns. The proofs are based on the Nullstellensatz and some elementary invariant-theoretic techniques.