Perfect Matchings, Rank of Connection Tensors and Graph Homomorphisms

TL;DR

This paper introduces a new algebraic framework using connection tensors to analyze graph properties, revealing limitations of graph homomorphism functions in expressing certain problems like perfect matchings.

Contribution

It generalizes the concept of connection matrices to connection tensors and establishes a simple exponential rank bound as a necessary and sufficient condition.

Findings

Counting perfect matchings cannot be expressed by graph homomorphism functions with complex weights.

Connection tensors generalize connection matrices for graph properties.

Positive semidefiniteness is not required in the more general setting.

Abstract

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.