  We develop a theory of graph algebras over general fields. This is modeled
after the theory developed by Freedman, Lov\'asz and Schrijver in [22] 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 with both complex vertex and edge
weights (or even from 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.
