TBI-Preprint 02-09-046
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Titel:
Graph Laplacians, Nodal Domains,
and Hyperplane Arrangements
Author(s):
Türker Biyikoglu,
Wim Hordijk,
Josef Leydold,
Tomaz Pisanski, and
Peter F. Stadler
Submitted to:
J. Exp. Math, (2002)
Abstract:
Eigenvectors of the Laplacian of a graph G have received increasing
attention in the recent past. Here we investigate their so-called nodal
domains, i.e., the connected components of the maximal induced subgraphs of
G on which an eigenvector ψ does not change sign. An analogue of
Courant's nodal domain theorem provides upper bounds on the number of nodal
domains depending on the location of ψ in the spectrum. This bound,
however, is not sharp in general. In this contribution we consider the
problem of computing minimal and maximal numbers of nodal domains for a
particular graph. The class of Boolean Hypercubes is discussed in detail.
We find that, despite the simplicity of this graph class, for which
complete spectral information is available, the computations are still
non-trivial. Nevertheless, we obtained some new results and a number of
conjectures.
Keywords:
Graph Laplacian, Hyperplane Arrangement, Nodal Domains, Landscapes
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Last modified: Mon Aug 19 18:23:01 CEST 2002