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We develop a theory of graph algebras over general fields. This is modeled after the theory developed by Freedman, Lovasz and Schrijver in [19] for connection matrices, in the study of graph homomorphism functions over real edge weight and positive vertex weight. We introduce connection tensors for graph properties. This notion naturally generalizes the concept of connection matrices. It is shown that counting perfect matchings, and a host of other graph properties naturally defined as Holant problems (edge models), cannot be expressed by graph homomorphism functions over the complex numbers (or even more general fields). Our necessary and sufficient condition in terms of connection tensors is a simple exponential rank bound. It shows that positive semidefiniteness is not needed in the more general setting.

Perfect Matchings, Rank of Connection Tensors and Graph Homomorphisms