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YCor
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Ferran V.
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Eigenfunctions adjacency operator on infinite graph in $l^2$

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?