The theory of designs concerns itself with questions about subsets of a set (or relations between two sets) possessing a high degree of symmetry. By contrast, the large and amorphous area called ‘graph theory’ is mainly concerned with questions about ‘general’ relations on a set. This generality means that usually either the questions answered are too particular, or the results obtained are not powerful enough, to have useful consequences for design theory. There are some places where the two theories have interacted fruitfully; in the next five chapters, several of these areas will be considered The unifying theme is provided by a class of graphs, the ‘strongly regular graphs’, introduced by Bose [11], whose definition reflects the symmetry inherent in t-designs. First, however, we shall mention an example of the kind of situation we shall not be discussing.

A graph consists of a finite set of vertices together with a set of edges, where each edge is a subset of the vertex set of cardinality 2. (In classical terms, our graphs are undirected, without loops or multiple edges.) As with designs, there is an alternative definition: a graph consists of a finite set of vertices and a set of edges, with an ‘incidence’ relation between vertices and edges, with the properties that any edge is incident with two vertices and any two vertices with at most one edge.