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.

chapter 1|18 pages

The Matchings Polynomial

chapter 2|18 pages

The Characteristic Polynomial

chapter 4|24 pages

Walk Generating Functions

chapter 5|18 pages

Quotients of Graphs

chapter 6|19 pages

Matchings and Walks

chapter 7|17 pages


chapter 8|17 pages

Orthogonal Polynomials

chapter 9|27 pages

Moment Sequences

chapter 10|17 pages

Strongly Regular Graphs

chapter 11|25 pages

Distance-Regular Graphs

chapter 12|39 pages

Association Schemes

chapter 13|23 pages

Representations of Distance-Regular Graphs

chapter 14|22 pages

Polynomial Spaces

chapter 15|25 pages

Q-Polynomial Spaces

chapter 16|19 pages

Tight Designs