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I think the answer is that every such graph should be quasi-isometric to the infinite line. Look at this paper. Theorem 9 there states that even a quasi-transivite amenable graph should have trivial Poisson boundary (no non-constant harmonic functions). On the other hand, a hyperbolic transitive graph, if it is not quasi-isometric to the Cayley graph $\mathbb Z$, always has non-trivial Poisson boundary. Here it is proved for groups, but I think it works for transitive hyperbolic graphs as well.