Eigenvectors of graph Laplacians have not, to date, been the
subject of expository articles and thus they may seem a surprising
topic for a book. The authors propose two motivations for this new
LNM volume: (1) There are fascinating subtle differences between
the properties of solutions of Schrodinger equations on manifolds
on the one hand, and their discrete analogs on graphs. (2)
Geometric properties of (cost) functions defined on the vertex sets
of graphs are of practical interest for heuristic optimization
algorithms. The observation that the cost functions of quite a few
of the well-studied combinatorial optimization problems are
eigenvectors of associated graph Laplacians has prompted the
investigation of such eigenvectors.
The volume investigates the structure of eigenvectors and looks
at the number of their sign graphs ( nodal domains ), Perron
components, graphs with extremal properties with respect to
eigenvectors. The Rayleigh quotient and rearrangement of graphs
form the main methodology."
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