Lecture Notes in Mathematics
cover

Laplacian Eigenvectors of Graphs

Perron-Frobenius and Faber-Krahn Type Theorems

Authors: Biyikoglu, Türker, Leydold, Josef, Stadler, Peter F.

  • A brand new take on a hitherto underreported area
  • Includes 35 illustrations
  • An insight into the authors’ cutting edge work
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eBook 29,74 €
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  • ISBN 978-3-540-73510-6
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Softcover 36,35 €
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About this book

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.

Reviews

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)


Table of contents (6 chapters)

Table of contents (6 chapters)
  • Introduction

    Pages 1-14

  • Graph Laplacians

    Pages 15-27

  • Eigenfunctions and Nodal Domains

    Pages 29-47

  • Nodal Domain Theorems for Special Graph Classes

    Pages 49-65

  • Computational Experiments

    Pages 67-75

Buy this book

eBook 29,74 €
price for Spain (gross)
  • ISBN 978-3-540-73510-6
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover 36,35 €
price for Spain (gross)
  • ISBN 978-3-540-73509-0
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
  • The final prices may differ from the prices shown due to specifics of VAT rules
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Bibliographic Information

Bibliographic Information
Book Title
Laplacian Eigenvectors of Graphs
Book Subtitle
Perron-Frobenius and Faber-Krahn Type Theorems
Authors
Series Title
Lecture Notes in Mathematics
Series Volume
1915
Copyright
2007
Publisher
Springer-Verlag Berlin Heidelberg
Copyright Holder
Springer-Verlag Berlin Heidelberg
eBook ISBN
978-3-540-73510-6
DOI
10.1007/978-3-540-73510-6
Softcover ISBN
978-3-540-73509-0
Series ISSN
0075-8434
Edition Number
1
Number of Pages
VIII, 120
Topics