Pavel Exner, Christian Seifert, Peter Stollmann
Absence of absolutely continuous spectrum for the Kirchhoff Laplacian on radial trees
(365K, pdf)

ABSTRACT.  In this paper we prove that the existence of absolutely continuous spectrum of the Kirchhoff Laplacian on a radial metric tree graph together with a finite complexity of the geometry of the tree implies that the tree is in fact eventually periodic. 
This complements the results by Breuer and Frank in