96-01-01.ley
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A Faber-Krahn-type inequality for regular trees

### Abstract

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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Josef.Leydold@statistik.wu-wien.ac.at