Reflection positivity, rank connectivity, and homomorphism of graphs
  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. 
M. Freedman (Microsoft Research), L. Lovasz (Microsoft Research), A.   Schrijver (CWI Amsterdam)