## Distance-Regular GraphsEver 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. |

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### Contents

SPECIAL REGULAR GRAPHS | 1 |

ASSOCIATION SCHEMES | 43 |

REPRESENTATION THEORY | 79 |

Copyright | |

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### Common terms and phrases

adjacent algebra amply regular antipodal antipodal covers association scheme automorphism group BANNAI BIGGs binary Golay code bound for i,j BROUweR classical parameters coclique common neighbours completely regular contains contradiction Corollary corresponding coset graph defined denote disjoint distance-regular graph distance-transitive graphs double coset dual polar graph edges eigenvalue equivalent finite follows forms graph geometry girth Golay code graph of diameter Grassmann graph Hadamard halved graphs Hamming graph hence Hoffman-Singleton graph implies imprimitive incidence graph induced inequality integer intersection array involutions isomorphic Johnson graph lattice Lemma line graph linear Math matrix maximal cliques multiplicity Odd graph partition permutation Petersen graph polygon polynomials precisely Proof Proposition Q-polynomial Q-sequence quadrangle Remark representation root shows singular lines smallest eigenvalue strongly regular graph subgraph subgroup subspace Suppose symmetric Terwilliger Theorem transitive two-graph unique valency vectors vertex set vertices