This graduate level text is distinguished both by the range of topics and the novelty of the material it treats--more than half of the material in it has previously only appeared in research papers. The first half of this book introduces the characteristic and matchings polynomials of a graph. It is instructive to consider these polynomials together because they have a number of properties in common. The matchings polynomial has links with a number of problems in combinatorial enumeration, particularly some of the current work on the combinatorics of orthogonal polynomials. This connection is discussed at some length, and is also in part the stimulus for the inclusion of chapters on orthogonal polynomials and formal power series. Many of the properties of orthogonal polynomials are derived from properties of characteristic polynomials. The second half of the book introduces the theory of polynomial spaces, which provide easy access to a number of important results in design theory, coding theory and the theory of association schemes. This book should be of interest to second year graduate text/reference in mathematics.
Table of Contents
1. The Matchings Polynomial 2. The Characteristic Polynomial 3. Formal Power Series and Generating Functions 4. Walk Generating Functions 5. Quotients of Graphs 6. Matchings and Walks 7. Pfaffians 8. Orthogonal Polynomials 9. Moment Sequences 10. Strongly Regular Graphs 11. Distance-Regular Graphs 12. Association Schemes 13. Representations of Distance-Regular Graphs 14. Polynomial Spaces 15. Q-Polynomial Spaces 16. Tight Designs
"The topics have been thoughtfully chosen and masterfully integrated into the text. The result is a delightfully entertaining and informative excursion into the field...this text provides the most comprehensive treatment I have encountered...this book is likely to become a permanent fixture in the study..."
- Mathematical Reviews