The Mutually Beneficial Relationship of Graphs and Matrices
About this Title
Richard A. Brualdi, University of Wisconsin, Madison, Madison, WI
Publication: CBMS Regional Conference Series in Mathematics
Publication Year 2011: Volume 115
ISBNs: 978-0-8218-5315-3 (print); 978-1-4704-1573-0 (online)
MathSciNet review: MR2808017
MSC: Primary 05C50; Secondary 05B35, 05C20, 15-02
Graphs and matrices enjoy a fascinating and mutually beneficial relationship. This interplay has benefited both graph theory and linear algebra. In one direction, knowledge about one of the graphs that can be associated with a matrix can be used to illuminate matrix properties and to get better information about the matrix. Examples include the use of digraphs to obtain strong results on diagonal dominance and eigenvalue inclusion regions and the use of the Rado-Hall theorem to deduce properties of special classes of matrices. Going the other way, linear algebraic properties of one of the matrices associated with a graph can be used to obtain useful combinatorial information about the graph. The adjacency matrix and the Laplacian matrix are two well-known matrices associated to a graph, and their eigenvalues encode important information about the graph. Another important linear algebraic invariant associated with a graph is the Colin de Verdière number, which, for instance, characterizes certain topological properties of the graph.
This book is not a comprehensive study of graphs and matrices. The particular content of the lectures was chosen for its accessibility, beauty, and current relevance, and for the possibility of enticing the audience to want to learn more.
Graduate students and research mathematicians interested in graph theory.
Table of Contents
- Chapter 1. Some fundamentals
- Chapter 2. Eigenvalues of graphs
- Chapter 3. Rado-Hall theorem and applications
- Chapter 4. Colin de Verdière number
- Chapter 5. Classes of matrices of zeros and ones
- Chapter 6. Matrix sign patterns
- Chapter 7. Eigenvalue inclusion and diagonal products
- Chapter 8. Tournaments
- Chapter 9. Two matrix polytopes
- Chapter 10. Digraphs and eigenvalues of (0,1)-matrices