By Jagdish Srivastava

ISBN-10: 0720422620

ISBN-13: 9780720422627

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Bridges and Ryser [1969] considered yet another generalization of block designs. Their («, k, 2)-systems on r and s are defined in terms of binary square matrices X and Y of order n satisfying the real matrix equation! XY = γχ= (k-X)I+XJ, k*k9 which, for λ Φ 0, implies JX = XJ = rJ, JY = YJ = sJ for integer r and s. In particular, («, k, 2)-systems on k and k 4-1 are character ized by symmetric matrices C with zero diagonal and elements + 1 and — 1 elsewhere, such that C 2 = (i+4(k-X))I+(n-2-4(k-X))J9 CJ = (2k-n+l)J.

Kb corresponding to the lines of the geometry satisfying the following axioms: A*l. Any two adjacent vertices of G are contained in one and only one clique of Σ. A*2. Each vertex of G is contained in r cliques of Σ. A*4. If P is a vertex of G not contained in a clique Kv of Σ, there are exactly / vertices of Kx which are adjacent to G. Conversely, if there exists a graph with a set of cliques Σ satisfying A*l-A*4, then it is isomorphic to a partial geometry (r, k, t), the vertices of the graph CH.

Then any two blocks of D intersect in exactly one treatment. The treatments and blocks of D can then be taken as the points and lines of a partial geometry (r, k, t), where r = k* and k = r*. If we call two treatments of D first associates or second associates according as they do or not occur together in the same block, we have a two class association scheme, and a corresponding strongly regular graph G with parameters v = fc[r(fc-l) + l]/r, 2 p } i = ( r - l ) + (fc-2), nx = r(fc-l), P!! = r 2 .