Four Amazing Positivities with Dimers/i-Matchings

2021/05/22 by Paul Federbush, Federbush, Paul
Engineering · Mathematics · #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Limits and Structures in Graph Theory #Mathematical Physics (math-ph) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2105.10772

openalex publication_date 2021/05/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We collect a number of striking recent results in a study of dimers on infinite regular bipartite lattices and also on regular bipartite graphs. We clearly separate rigorously proven results from conjectures. A primary goal is to show people: here is a field which is ripe for further interesting research. We separate four classes of endeavor, of which we here extract two items to whet one's appetite. Primo,for hyper-rectangular lattices of every dimension the first 20 virial coefficients are positive. (One has no understanding of this yet!) Secondo, all regular bipartite graphs with less than 14 vertices satisfy graph positivity, defined below. (Here there is some understanding.)

Citations

• A PROOF of Weak Graph Positivity, for a Large Range of the Parameters

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