The theory of association schemes arose in statistics, in the theory of experimental design for the analysis of variance. In mathematics, association schemes belong to both algebra and combinatorics. In algebraic combinatorics, association schemes provide a unified approach to many topics, for example combinatorial designs and coding theory. In algebra, association schemes generalize groups, and the theory of association schemes generalizes the character theory of linear representations of groups. Definition An n-class association scheme consists of a set X together with a partition S of X × X into n + 1 binary relations, R0, R1, ..., Rn which satisfy: and is called the identity relation. Defining , if R in S, then R* in S If , the number of such that and is a constant depending on , , but not on the particular choice of and . An association scheme is commutative if for all , and . Most authors assume this property. A symmetric association scheme is one in which each is a symmetric relation. That is: if (x, y) ∈ Ri, then (y, x) ∈ Ri. (Or equivalently, R* = R.) Every symmetric association scheme is commutative. Note, however, that while the notion of an association scheme generalizes the notion of a group, the notion of a commutative association scheme only generalizes the notion of a commutative group. Two points x and y are called i th associates if . The definition states that if x and y are i th associates then so are y and x. Every pair of points are i th associates for exactly one . Each point is its own zeroth associate while distinct points are never zeroth associates. If x and y are k th associates then the number of points which are both i th associates of and j th associates of is a constant . Graph interpretation and adjacency matrices A symmetric association scheme can be visualized as a complete graph with labeled edges. The graph has vertices, one for each point of , and the edge joining vertices and is labeled if and are th associates. Each edge has a unique label, and the number of triangles with a fixed base labeled having the other edges labeled and is a constant , depending on but not on the choice of the base. In particular, each vertex is incident with exactly edges labeled ; is the valency