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Three important properties of groups, amenability, Property A and almostfiniteness, are in our focus, in the wider context of general countable boundeddegree graphs. A graph is almost finite if it has a tiling with isomorphiccopies of finitely many F\o lner sets, and we call it strongly almost finite,if the tiling can be randomized so that the probability that a vertex is on theboundary of a tile is uniformly small. We give various equivalents for PropertyA and for strong almost finiteness. In particular, we prove that Property Atogether with a uniform version of amenability is equivalent to strong almostfiniteness. Using these characterizations, we show that graphs ofsubexponential growth and Schreier graphs of amenable groups are alwaysstrongly almost finite, generalizing the celebrated result of Downarowicz,Huczek and Zhang about amenable Cayley graphs, based on graph theoretic ratherthan group theoretic principles. We also show that if a sequence of graphs ofProperty A (in a uniform sense) converges to a graph $G$ in the neighborhooddistance (a purely combinatorial analogue of the classical Benjamini-Schrammdistance), then their Laplacian spectra converge to the Laplacian spectrum of$G$ in the Hausdorff distance. Finally, we apply the previous theory toconstruct a new and rich class of classifiable $C^{\star}$-algebras. Namely, weshow that for any minimal strongly almost finite graph $G$ there are naturallyassociated simple, nuclear, stably finite $C^{\star}$-algebras that areclassifiable by their Elliott invariants.

A comprehensive characterization of Property A and almost finiteness