Edge-reflection positivity and weighted graph homomorphisms
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.
Keywords
Cite
@article{arxiv.1302.6497,
title = {Edge-reflection positivity and weighted graph homomorphisms},
author = {Guus Regts},
journal= {arXiv preprint arXiv:1302.6497},
year = {2014}
}
Comments
Some typos and inconsistencies have been fixed. !0 pages. To appear in Journal of Combinatorial Series A