A Faber-Krahn-type inequality for regular trees

Josef Leydold


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 or Cheeger's inequality. Friedman (Some geometric aspects of graphs and their eigenfunctions, Duke Math. J. 69 (3), pp. 487-525, 1993) described the idea of a ``graph with boundary''. With this concept it is possible to formulate Dirichlet and Neumann eigenvalue problems. Friedman also conjectured another ``classical'' result for manifolds, the Faber-Krahn theorem, for regular bounded trees with boundary. The Faber-Krahn theorem states that among all bounded domains $D \subset R^n$ with fixed volume, a ball has lowest first Dirichlet eigenvalue.

In this paper we show such a result for regular trees by using a rearrangement technique. We give restrictive conditions for trees with boundary where the first Dirichlet eigenvalue is minimized for a given ``volume''. Amazingly Friedman's conjecture is false, i.e. in general these trees are not ``balls''. But we will show that these are similar to ``balls''.

Mathematics Subject Classification: 58-99 (Global analysis, analysis on manifolds), 05C99 (Graph theory)

Key Words: regular tree, graph laplacian, Dirichlet eigenvalue problem, Faber-Krahn inequality, first eigenvalue, eigenfunction, rearrangment

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