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**Example text**

Cs with the properties (i) given i E { 1, ... , s } and x E X, the number ni of y E X with {x, y) E C . , S) and x, y E X with ix, y } EC W the number aijk of z E X with {x, z I E Ci and {y, z) E Cj depends only on i, j, k (and not on x, y). Here x and y are sometimes said to be ith associates if ix, Y J EC i - hence the name. Alternatively we can regard Ci as the set of edges of a graph, distinguishing the different graphs where necessary by colouring their edges with different colours; then in particular each monochrome subgraph is regular.

An application will be given in Theorem 5. 12. There are many others, for example Hoffman and Singleton [18]. 40 Note that there is an irreducible representation (which we may as well take to be p0) given by p0(Ai) = ni for all i; a corresponding eigenvector is the all-1 vector, since n i is the row and column sum of Ai The multiplicity is f 0 = 1. We now turn to a restricted class of association schemes. In a connected graph r with vertex set X, the distance d(x, y) from x to y is the length of (or number of edges in) the shortest path joining x and y; this function is a metric on X in the usual sense.

There are (t + 2) elements of (t P 2); of these, (t) q + 2) /(t) are subsets of blocks, and (t) (t + 1) (v - k) /(t) have the property that some block contains all but one of them. : (;*t + 2) + (t k 1)(° - k)], which reduces to 1)(k 1)) Theorems 3A. 4 and 3A. 5 both give lower bounds for v in terms of k and t. The first is stronger for k > 2(t - 1), the second for k < 2(t - 1), while if k = 2(t - 1) then both assert that v ? t2 - 1. Note that 3A. 5, but not 3A. 4, bounds the number of times a system may be extended.