By Andries E. Brouwer, Arjeh M. Cohen, Arnold Neumaier (auth.)

ISBN-10: 3642743412

ISBN-13: 9783642743412

ISBN-10: 3642743439

ISBN-13: 9783642743436

Ever because the discovery of the 5 platonic solids in precedent days, the examine of symmetry and regularity has been some of the most attention-grabbing points of arithmetic. regularly the arithmetical regularity houses of an item suggest its distinctiveness and the life of many symmetries. This interaction among regularity and symmetry houses of graphs is the subject matter of this ebook. ranging from very basic regularity houses, the concept that of a distance-regular graph arises clearly as a typical environment for normal graphs that are extremal in a single experience or one other. numerous different very important general combinatorial constructions are then proven to be such as distinct households of distance-regular graphs. different matters of extra normal curiosity, equivalent to regularity and extremal houses in graphs, organization schemes, representations of graphs in euclidean area, teams and geometries of Lie variety, teams performing on graphs, and codes are coated independently. Many new effects and proofs and greater than 750 references raise the encyclopaedic price of this book.

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Special Regular Graphs precisely n /4 places. Thus, either n Eo;; 2 or n is divisible by 4. The case n Eo;; 2 is trivial. It is a long-standing conjecture that Hadamard matrices exist for all orders n 0 (mod 4). , for n = 4m < 428 and for n = 23 m with m 2 Eo;; '):'; see WALLIS, STREET & WALLIS [768], SEBERRY [767], GERAMITA & SEBERRY [322] and the survey article of HEDAYAT & WALLIS [367] for the many known construction techniques and SAWADE [637] for the current state of knowledge. 1. Theorem (DRAKE [271], SHAD [662], DELORME [246]).

This result and the well-known classification of all finite generalized quadrangles with line size 3 is the content of the theorem below. Recall that a generalized quadrangle is a partial linear space such that if x is a point not on a line I, then x is collinear with a unique point of I. It is called degenerate if some point is collinear with all other points. Thus, in a degenerate generalized quadrangle the set of lines forms a pencil. Write Lx for the set of lines on x. One easily sees that if x and y are noncollinear, then 1Lx 1 = 1Ly I.

A bipartite graph with v vertices is distance-regular of diameter d = 3 (with intersection array {k, k -1, k - 1-'; 1,1-', k}) if and only if it is the incidence graph of a square 2-design (with parameters 2-(w,k,I-'), where k > 1 and w = = I+k(k-I)/I-'). tv Proof. Let X I and X 2 be the parts of a bipartite distance-regular graph f of diameter 3. Interpret the adjacency relation of f as incidence on Xl XX 2 • It is obvious that this defines a square 2-design with the stated parameters. The converse also holds as any two blocks of a square 2-(w,k, 1-') design meet in exactly I-' points of the design.

### Distance-Regular Graphs by Andries E. Brouwer, Arjeh M. Cohen, Arnold Neumaier (auth.)

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