Perfect matchings, rank of connection tensors and graph homomorphisms

Abstract We develop a theory of graph algebras over general fields. This is modelled after the developed by Freedman et al. (2007, J. Amer. Math. Soc. 20 37–51) for connection matrices, in study homomorphism functions real edge weight and positive vertex weight. introduce tensors properties. notion naturally generalizes concept matrices. It shown that counting perfect matchings, host other properties defined as Holant problems (edge models), cannot be expressed with both complex weights (or even from more fields). Our necessary sufficient condition terms simple exponential rank bound. shows semidefiniteness not needed setting.