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Brouwer A.E., Cohen A.M., Neumaier A. Distance Regular Graphs

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Brouwer A.E., Cohen A.M., Neumaier A. Distance Regular Graphs
Издательство Springer, 1989, -509 pp.
Graphs are studied in a wide variety of contexts. This is due to the fact that the con- cept of a graph is both general and convenient. It is convenient because mathematical as well as intuitive notions can often be formulated very easily in terms of symmetric relations; it is general because there are so many symmetric relations on a set. In par- ticular, there is no hope for the classification of all finite graphs in the vein of the classification of, say, all finite fields. However, the situation may change if regularity or symmetry is assumed to exist.
Since the ancient determination of the five platonic solids, the study of symmetry and regularity has always been one of the most fascinating aspects of mathematics, and even now there are many challenging problems in this area. One intriguing phenomenon is the fact that quite often arithmetical regularity properties of an object imply the uniqueness of the object, and (often as a consequence of the uniqueness proof) imply the existence of many symmetries, i.e., large automorphism groups. This interplay between regularity and symmetry properties of graphs is the theme of this book.
Many regularity properties are naturally expressible in terms of an association scheme, which is by far the most important unifying concept in algebraic combinatorics. Of special interest are two classes of association schemes, those which are polynomial or Q-polynomial. The P-polynomial association schemes (i.e., those with a linear distribution diagram with respect to some relation) are essentially the same objects as distance-regular graphs. As we shall see, the theory of distance-regular graphs has connections to many parts of graph theory, design theory, coding theory, geometry (both finite and Euclidean), and group theory. Indeed, most finite objects of sufficient regularity are closely related to certain distance-regular graphs.
Special Regular Graphs
Association Schemes
Representation Theory
Theory of Distance-Regular Graphs
Parameter Restrictions for Distance-Regular Graphs
Classification of the Known Distance-Regular Graphs
Distance-Transitive Graphs
Q-polynomial Distance-Regular Graphs
The Families of Graphs with Classical Parameters
Graphs of Coxeter and Lie Type
Graphs Related to Codes
Graphs Related to Classical Geometries
Sporadic Graphs
Tables of Parameters for Distance-Regular Graphs
Appendix
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