Edge Coloring Models and Reflection Positivity

The motivation of this paper comes from statistical physics as well as from combinatorics and topology. The general setup in statistical mechanics can be outlined as follows. Let G be a graph and let C be a finite set of “states” or “colors”. We think of G as a crystal in which either the edges or the vertices are regarded as “sites” which can have states from C. In the first case we speak about edge coloring models and in the second case about vertex coloring models. A configuration of the whole system is a function which associates a state with each site. The states are interacting with each other at the vertices in edge coloring models and along edges in vertex coloring models. A weight is associated with each such interaction which is a real (or complex) number depending on the interacting states (in vertex coloring models there are additional weights associated with the states). A concrete model is usually given by these numbers. The partition function can be interpreted as a graph parameter which is computed by summing the products of the weights over all possible configurations of the system represented by G. It proves to be useful to extend this graph parameter linearly to the vector space of formal linear combinations of graphs. The elements of this vector space are called quantum graphs. Quantum graphs that can be obtained by gluing together a quantum graph with its reflected version (using the distributive law) are called reflection symmetric. However there are two different reasonable definitions of gluing. In the first one we glue along unfinished edges and in the second one along vertices. Correspondingly we get the notions of edge reflection symmetric and vertex reflection symmetric quantum graphs. A graph parameter is called edge reflection positive (resp. vertex reflection positive) if it takes nonnegative values on edge-reflection symmetric (resp. vertex reflection symmetric) quantum graphs. It is a simple fact that the partition function in edge coloring models is edge reflection positive and is vertex reflection positive in vertex coloring models. A surprising result proved by M. H. Freedman, L. Lovász and A. Schrijver (see [4]) says that vertex reflection positivity is almost enough to characterize the partition functions of vertex coloring models. The extra condition that they need is that the ranks of certain matrices (which describe the gluing operation and are called connection matrices) are growing at most exponentially. They conjectured that similar characterization can be given for edge reflection positive graph parameters. The main result of this paper (theorem 2.2) is the proof of this conjecture in a strong version where we replace the condition on the rank growth by a weak and