Spectral Graph Drawing

✍ Scribed by Thomas Puppe

• Year: 2008
• Tongue: English
• Leaves: 96
• Category: Library

✦ Synopsis

Graph Drawing is the science of finding an intuitive visualization of a network (or in mathematical terms of a graph). One approach is to define energy functions that represent design criteria for graph layouts. It happens to be that the eigenvalues of graph related matrices are locally optimal solutions for some of the energy functions. Using the eigenvalues for a graph layout is called Spectral Graph Drawing.This book is a survey of Spectral Graph Drawing methods. Graph layouts of several graph-related matrices, such as the adjacency or the Laplace matrix, are studied. There is a special section on the implementation of the graph layouts using the power iteration. At the end the focus is extended to the special requirements for Dynamic Spectral Graph Drawing, i.e. time-variant graphs are drawn with spectral methods.

✦ Table of Contents

Zusammenfassung......Page 6
Introduction......Page 8
Graph Theory......Page 10
Basic Definitions......Page 11
Eigentheory......Page 13
Real Symmetric Matrices......Page 14
The Generalized Eigenvalue Problem......Page 17
Gershgorin's Discs and Extensions......Page 20
Perturbation Theory......Page 22
Adjacency Matrix......Page 26
Degree Matrix......Page 27
Laplace Matrix......Page 29
Relaxed Laplace Matrix......Page 31
Generalized Laplace Matrix......Page 35
Normalized Laplace Matrix......Page 37
Isomorphisms......Page 39
Eigenvalue Bounds......Page 40
Bounds of the Relaxed Laplace Matrix......Page 41
Bounds of the Generalized Laplace Matrix......Page 47
Motivation......Page 50
Laplace Layout......Page 53
Relaxed Laplace Layout......Page 57
Generalized Laplace Layout......Page 64
A Spectral Layout Algorithm......Page 68
Convergence Anormalities......Page 78
Dynamic Graph Drawing Using Spectral Layouts......Page 85
Conclusion......Page 91
Content of the Enclosed CD......Page 93