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.
The vertex connection matrices were used by Freedman, Lov 'asz and Schrijver in @cite_4 to characterize graph parameters which are of the form @math for some positive @math and symmetric @math .
@cite_4
1975568166 2003078549 2068332510 2141382791
Connection matrices were introduced in [M. Freedman, L. Lovasz, 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. 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. It is well known that the smallest eigenvalue of the adjacency matrix of a connected d-regular graph is at least − d and is strictly greater than − d if the graph is not bipartite. More generally, for any connected graph G = (V, E), consider the matrix Q = D + A where D is the diagonal matrix of degrees in the graph G and A is the adjacency matrix of G. Then Q is positive semidefinite, and the smallest eigenvalue of Q is 0 if and only if G is bipartite. We will study the separation of this eigenvalue from 0 in terms of the following measure of nonbipartiteness of G. For any S ⊆ V, we denote by emin(S) the minimum number of edges that need to be removed from the induced subgraph on S to make it bipartite. Also, we denote by cut(S) the set of edges with one end in S and the other in V − S. We define the parameter Ψ as ***image*** The parameter Ψ is a measure of the nonbipartiteness of the graph G. We will show that the smallest eigenvalue of Q is bounded above and below by functions of Ψ. For d-regular graphs, this characterizes the separation of the smallest eigenvalue of the adjacency matrix from −d. These results can be easily extended to weighted graphs. © 1994 Wiley Periodicals, Inc. It is shown that a graph parameter can be realized as the number of homomorphisms into a fixed (weighted) graph if and only if it satisfies two linear algebraic conditions: reflection positivity and exponential rank-connectivity. In terms of statistical physics, this can be viewed as a characterization of partition functions of vertex models.