  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.
