Edge-reflection positivity and weighted graph homomorphisms

TL;DR

This paper characterizes when the number of graph homomorphisms can be represented by real-valued edge-coloring models, linking reflection positivity with geometric invariant theory, and explicitly identifies simple graphs with this property.

Contribution

It provides a geometric invariant theory-based characterization of weighted graphs with real-valued edge-coloring models, answering Szegedy's question about reflection positivity.

Findings

Identifies conditions for real-valued edge-coloring models to represent homomorphism counts.

Explicitly characterizes simple graphs with edge-reflection positive homomorphism counts.

Connects reflection positivity with geometric invariant theory in graph models.

Abstract

B. Szegedy [Edge coloring models and reflection positivity, {\sl Journal of the American Mathematical Society} {\bf 20} (2007) 969--988] showed that the number of homomorphisms into a weighted graph is equal to the partition function of a complex edge-coloring model. Using some results in geometric invariant theory, we characterize for which weighted graphs the edge-coloring model can be taken to be real valued that is, we characterize for which weighted graphs the number of homomorphisms into them are edge-reflection positive. In particular, we determine explicitly for which simple graphs the number of homomorphisms into them is equal to the partition function of a real edge-coloring model. This answers a question posed by Szegedy.