Graphs and Geometry

Description

Graphs are usually represented as geometric objects drawn in the plane, consisting of nodes and curves connecting them. The main message of this book is that such a representation is not merely a way to visualize the graph, but an important mathematical tool. It is obvious that this geometry is crucial in engineering, for example, if you want to understand rigidity of frameworks and mobility of mechanisms. But even if there is no geometry directly connected to the graph-theoretic problem, a well-chosen geometric embedding has mathematical meaning and applications in proofs and algorithms. This book surveys a number of such connections between graph theory and geometry: among others, rubber band representations, coin representations, orthogonal representations, and discrete analytic functions. Applications are given in information theory, statistical physics, graph algorithms and quantum physics. The book is based on courses and lectures that the author has given over the last few decades and offers readers with some knowledge of graph theory, linear algebra, and probability a thorough introduction to this exciting new area with a large collection of illuminating examples and exercises. Dedication Contents Preface 1. Why are Geometric Representations Interesting? 2. Planar Graphs 3. Rubber Bands 4. Discrete Harmonic Functions 5. Coin Representation 6. Square Tilings 7. Discrete Analytic Functions 8. Discrete Analytic Functions: Statistical Physics 9. Adjacency Matrix and its Square 10. Orthogonal Representations: Dimension 11. Orthogonal Representations: The Smallest Cone 12. Orthogonal Representations: Quantum Physics 13. Semidefinite Optimization 14. Stresses 15. Rigidity and Motions of Frameworks 16. The Colin de Verdière Number 17. Metric Representations 18. Matching and Covering in Frameworks 19. Combinatorics of Subspaces 20. Concluding Thoughts Appendix A. Linear Algebra Appendix B. Graphs Appendix C. Convex Bodies Bibliography Author Index Subject Index