Theory of Distance-Regular Graphs

Abstract

We now come to the central topic of the book. The first section may be viewed as a short introduction to the subject. Although we shall develop large parts of the theory of distance-regular graphs independently of Chapter 2, we shall use concepts and results about association schemes for more specialized topics such as Q-polynomial orderings (Chapter 8) and codes in graphs (Chapter 11). In §4.2 we look at various constructions that, given a distance-regular graph, produce a new one. In §4.3 we show how certain conditions on the parameters force the presence of substructures, like lines or Petersen subgraphs. In §4.4 we use the results of Chapter 3 to obtain a characterization by parameters of the two most basic families of distance-regular graphs, the Johnson and Hamming graphs. Chapter 5 contains most of the known conditions on the parameters, Chapter 6 classifies the known distance-regular graphs in various families, Chapter 7 is concerned with distance-transitive graphs, Chapter 8 discusses the consequences of the Q-polynomial property, and the remaining chapters give all examples known to us.