Distance-Regular Graphs (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics (18))

Ever since the discovery of the five platonic solids in ancient times, the study of symmetry and regularity has been one of the most fascinating aspects of mathematics. Quite often the arithmetical regularity properties of an object imply its uniqueness and the existence of many symmetries. This interplay between regularity and symmetry properties of graphs is the theme of this book. Starting from very elementary regularity properties, the concept of a distance-regular graph arises naturally as a common setting for regular graphs which are extremal in one sense or another. Several other important regular combinatorial structures are then shown to be equivalent to special families of distance-regular graphs. Other subjects of more general interest, such as regularity and extremal properties in graphs, association schemes, representations of graphs in euclidean space, groups and geometries of Lie type, groups acting on graphs, and codes are covered independently. Many new results and proofs and more than 750 references increase the encyclopaedic value of this book. Front Matter....Pages i-xvii Special Regular Graphs....Pages 1-42 Association Schemes....Pages 43-78 Representation Theory....Pages 79-125 Theory of Distance-Regular Graphs....Pages 126-166 Parameter Restrictions for Distance-Regular Graphs....Pages 167-192 Classification of the Known Distance-Regular Graphs....Pages 193-213 Distance-Transitive Graphs....Pages 214-234 Q-polynomial Distance-Regular Graphs....Pages 235-254 The Families of Graphs with Classical Parameters....Pages 255-293 Graphs of Coxeter and Lie Type....Pages 294-344 Graphs Related to Codes....Pages 345-373 Graphs Related to Classical Geometries....Pages 374-390 Sporadic Graphs....Pages 391-412 Tables of Parameters for Distance Regular Graphs....Pages 413-432 Back Matter....Pages 433-495 A.e. Brouwer, A.m. Cohen, A. Neumaier. Includes Bibliographical References And Indexes.