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

by Türker Biyikoglu, Josef Leydold, Peter F. Stadler

Overview

This fascinating volume investigates the structure of eigenvectors and looks at the number of their sign graphs ("nodal domains"), Perron components, and graphs with extremal properties with respect to eigenvectors. The Rayleigh quotient and rearrangement of graphs form the main methodology. Eigenvectors of graph Laplacians may seem a surprising topic for a book, but the authors show that there are subtle differences between the properties of solutions of Schrödinger equations on manifolds on the one hand, and their discrete analogs on graphs.

Product Details

ISBN-13:	9783540735090
Publisher:	Springer Berlin Heidelberg
Publication date:	09/10/2007
Series:	Lecture Notes in Mathematics , #1915
Edition description:	2007
Pages:	120
Product dimensions:	6.10(w) x 9.25(h) x 0.01(d)

Table of Contents

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

Editorial 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)

From the Publisher