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.
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SPECIAL REGULAR GRAPHS
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absolute bound adjacent algebra amply regular antipodal antipodal covers association scheme automorphism group Bannai Biggs binary Golay code Brouwer classical parameters coclique common neighbours completely regular contains contradiction Corollary corresponding Coxeter graph defined denote disjoint distance-regular graph distance-transitive graphs double coset dual polar graph edges eigenvalue equality in absolute equivalent finite follows forms graph geometry girth graph of diameter graph with intersection Grassmann graph Hadamard halved graphs Hamming graph hence Hoffman-Singleton graph implies imprimitive incidence graph induced inner product integer intersection array involutions isomorphic Johnson graph lattice Lemma line graph linear Math matrix maximal cliques multiplicity Neumaier Odd graph orbits partition permutation Petersen graph polygon polynomial precisely Proof Proposition Q-polynomial quadrangle Remark representation shows singular lines smallest eigenvalue strongly regular graph subgraph subgroup subset subspace Suppose symmetric T2 is strongly Terwilliger Theorem transitive two-graph unique valency vectors vertex set