We develop a theory of graph algebras over general fields. This is modelled after the theory developed by Freedman et al. (2007, J. Amer. Math. Soc. 20 37–51) 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.

我们在一般领域发展了图代数理论。这是在 Freedman 等人开发的理论之后建模的。（2007 年，J. Amer. Math. Soc. 2037-51）对于连接矩阵，用于研究图同态函数超过实边权重和正顶点权重。我们为图属性引入连接张量。这个概念自然地概括了连接矩阵的概念。结果表明，计算完美匹配以及许多其他自然定义为 Holant 问题（边模型）的图属性，不能用具有复杂顶点和边权重的图同态函数（甚至更一般的领域）来表示。就连接张量而言，我们的充要条件是一个简单的指数秩界限。它表明在更一般的情况下不需要正半定性。