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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.

Edit. As R W says, the first reference in my answer is not relevant. It is nice that the statement is still true, and that it an be proved using Poisson boundaries.

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.

Edit. As R W says, the first reference in my answer is not relevant. It is nice that the statement is still true, and that it can be proved using Poisson boundaries.

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.

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.

Edit. As R W says, the first reference in my answer is not relevant. It is nice that the statement is still true, and that it an be proved using Poisson boundaries.

In general, if you take any graph, and add an infinite line to it (say, intersecting the original graph by a vertex), you get an amenable graph. If the original graph was hyperbolic, the new one will be hyperbolic too.

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.