Eigenfunctions adjacency operator on infinite graph in $l^2$

Question

Let $\Gamma$ be an infinite (connected) graph without edges going from a vertex to itself (though it might have multi-edges). Let us suppose that $\Gamma$ has finite valence.

Is there always a positive eigenfunction for the adjacency operator on $\Gamma$ which lives in $l^2$?

If not, under what conditions can we guarantee the existence of such an eigenfunction?

Answer 1

Not at all, the simplest counterexample is the integer line. Generally, the eigenvalue of any positive square integrable eigenfunction must coincide with the spectral radius, and existence of such functions is a pretty "rare" phenomenon (cf. Sullivan's paper Related aspects of positivity in Riemannian geometry in the setup of Riemannian manifolds).