The geometry of regular trees with the Faber?Krahn property

In this paper we prove a Faber-Krahn-type inequality for regular trees and give a complete characterization of extremal trees. The main tools are rearrangements and perturbation of regular trees. 1. Introduction In the last years some results for the Laplacian on manifolds have been shown to hold also for the graph Laplacian, e.g. Courant's nodal domain theorem ((dV93, Fri93]) or Cheeger's inequality ((dV94]). In Fri93] Friedman described the idea of a \graph with boundary" (see below). With this concept he was able to formulate Dirichlet and Neumann eigenvalue problems. He also conjectured another \clas-sical" result for manifolds, the Faber-Krahn theorem, for regular bounded trees with boundary. The Faber-Krahn theorem states that among all bounded domains