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arXiv · 2005.12107

Atoms of the matching measure

Ferenc Bencs·András Mészáros
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Abstract

We prove that the matching measure of an infinite vertex-transitive connected graph has no atoms. Generalizing the results of Salez, we show that for an ergodic non-amenable unimodular random rooted graph with uniformly bounded degrees, the matching measure has only finitely many atoms. Ku and Chen proved the analogue of the Gallai-Edmonds structure theorem for non-zero roots of the matching polynomial for finite graphs. We extend their results for infinite graphs. We also show that the corresponding Gallai-Edmonds decomposition is compatible with the zero temperature monomer-dimer model.

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Ferenc Bencs, András Mészáros. 2020-05-25. Atoms of the matching measure. https://arxiv.org/abs/2005.12107

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