Laplacian Eigenvectors of Graphs: Perron-Frobenius and Faber-Krahn Type Theorems

ISBN-13: 9783540735090 / Angielski / MiÄ™kka / 2007 / 128 str.

Turker Biyikoglu; Josef Leydold; Peter F. Stadler

Opis

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."

Recenzje

From the reviews:

"This is an interesting ... book about a very specialized topic in spectral graph theory, namely the eigenvectors of the (generalized) Laplacian matrices of graphs. ... Overall I found the book well worth reading, with a clear and novel presentation of some interesting ideas." (Gordon F. Royle, Mathematical Reviews, Issue 2009 a)

"The present book covers a narrow topic on the border of graph theory, geometry and analysis. ... The intended readership is broad. ... The presentation is ... clear. It is certainly stimulating and well worth to read for graduate students or researchers who encounter eigenvectors connected to discrete or continuous/geometrical structures." (Péter Hajnal, Acta Scientiarum Mathematicarum, Vol. 75, 2009)

Spis treści

Graph Laplacians.- Eigenfunctions and Nodal Domains.- Nodal Domain Theorems for Special Graph Classes.- Computational Experiments.- Faber-Krahn Type Inequalities.

Blurb

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 Schrödinger 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.