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I was wondering whether there exists any infinite tree $T$ such that the action of $\mathit{Aut}(T)$ on the set of vertices $V=V(T)$ has finitely many orbits, and whose spectrum $\sigma(T)$ has strictly more than 3 connected components.

I have tried and looked in the literature, but I couldn't find any example, nor a proof of non-existence.

For the clarity, here I am interested in the spectrum of the adjacency operator acting on $l^2(V)$.

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  • $\begingroup$ To get a start on this, can you name graphs whose spectrum has exactly 1,2 or 3 connected components, and a graph with more than 3 conneted components but infinitely many orbits on the vertices? $\endgroup$ – M. Winter Jan 8 at 13:44
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    $\begingroup$ @M.Winter The spectrum of the regular tree has 1 connected component, the spectrum of a semi-regular tree has 3, and the spectrum of an infinite path with a pendant vertex attach to each of the nodes in the path has 2. I have not looked much into the case with infinitely many orbits yet, for the moment I am more interested in the case of finitely many orbits. $\endgroup$ – Maurizio Moreschi Jan 13 at 7:52

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