Perfect Matchings, Rank of Connection Tensors and Graph Homomorphisms

Abstract

We develop a theory of graph algebras over general fields. This is modeled after the theory developed by Freedman, Lovász 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. Artem Govorov is the author's preferred spelling of his name, rather than the official spelling Artsiom Hovarau. * Balázs Szegedy [29] studied "edge coloring models" which are equivalent to a special case of Holant problems where at each vertex of degree d a single symmetric function of arity d is provided. In general Holant problems allow different (possibly non-symmetric) constraint functions from a set assigned at vertices; see [5].