# Read e-book online Combinatorial Mathematics, Optimal Designs and Their PDF By J. Srivastava (Eds.)

ISBN-10: 0444860487

ISBN-13: 9780444860484

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X,-,. Then = C:= c I G I ~ (f(a))*/x, =in'+ aeG c (n--i)rc,)/X,) ISL (here c, = I{a E G :f(a)= n - i}l) with equality iff G i s t-transitive. It is well-known also that G has r = (Caec f(a))/lGI orbits. It proves directly that lA*(t, n)l s n - t (r = (r+(IG)- I)t)/JGJ, IG1= ( n - t)/(r- t ) n ~- t ) and A*(t, (GI= n,! n,! . n,! where n = n , + . * + n, is a partition of n by lengths of orbits; so the maximum of G corresponds to n = 1+ .

Let m be the multiplicity of pl. 2), + p2 = ( v - 1)-'{-2t + ( v - 1- 2m)mp'(v - 1- m)-'[vmt(v - 1 - m ) ( u- 1- t);}. This is an integer only if m = i ( v - 1) or v m t ( v - 1 - m ) ( v - 1- t ) is a perfect square. This gives a necessary condition for the existence of a strongly regular graph in terms of the number of vertices, degree, and the multiplicity of the bigger eigenvalue. Since both of p1+ p2 and p1p2 are integers, it can be easily seen that either both of p1and p2 are integers or they are conjugate to each other.

It is called the Hamming scheme. Some subsets of a Hamming scheme are also schemes with the same definitions of distance and i-association (for example, the set of all binary u-vectors, each with the same number of ones). Now we will give an example of such subschemes of a Hamming scheme consisting only of permutations of (0, 1, . . , q - l}. e. n is not a prime. For any n,-vector x = (x,, . . , xn2)over {0,1,. . , n , - l},we define the following permutation of 1, . . , n i - l } : ( 1, . .