Atoms of the matching measure

@article{Bencs2020AtomsOT,
  title={Atoms of the matching measure},
  author={Ferenc Bencs and Andr'as M'esz'aros},
  journal={Electronic Journal of Probability},
  year={2020}
}
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… 
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A refined Gallai-Edmonds structure theorem for weighted matching polynomials

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A graph G for purposes here is a finite set of elements called vertices and a finite set of elements called edges such that each edge meets exactly two vertices, called the end-points of the edge. An…