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.

In a graph, a path is a sequence of vertices in which consecutive vertices are adjacent, and a circuit is a path with initial and terminal point equal. A graph is connected if any two vertices lie in a path. The function d defined by d(p, q) = length of the shortest path containing p and q, is a metric in a connected graph; the diameter of the graph is the largest value assumed by d. The girth of a graph is the length of the shortest circuit which contains no repeated edges, provided such a circuit exists.

For strongly regular graphs, the connectedness, diameter and girth are simply determined by the parameters. Γ is connected with diameter 2 if d > 0, and is disconnected if d = 0. Γ has a girth provided a > 1; the girth is 3 if c > 0, 4 if c = 0 and d > 1, and 5 if c = 0, d = 1.

It is easy to see that a graph with diameter 2 and maximal valency a has at most a2 + 1 vertices; and a graph with girth 5 and minimal valency a has at least a2 + 1 vertices. Equality holds in either case if and only if the graph is strongly regular with c = 0, d = 1. Such a graph is called a Moore graph with diameter 2.

In the year 1973, Professor van Lint and Dr. Cameron visited Westfield College separately on various occasions to lecture at our Seminar on Combinatorial Algebra and Geometry. As the subject matter of their lectures overlapped in a most interesting way, the idea was conceived of organizing the notes already prepared by each of them into a coherent whole. The present volume is the outcome.

The object of the lectures, and of these lecture notes, was to present to an audience already familiar with the theory of designs some of the connections and applications with other branches of mathematics. (For the present book, however, a short introductory chapter on designs has been included.) In particular, this has meant largely graphs and codes, and to both of these subjects as well some kind of introduction is given. But the object has been to stress the connections of design theory with these other areas, and so no attempt at a consistent coverage of graph theory and coding theory has been made.

The perceptive reader may well detect a difference in style between various chapters, revealing the particular approaches of the two authors. But we believe that the overall mathematical unity of the lectures and of the notes will be evident, and very useful to students and research workers in these fields.

In this section we shall be considering graphs from a slightly different point of view, as incidence structures in their own right (as remarked in Chapter 2). In particular, a regular graph is a 1-design with k = 2, and conversely (provided we forbid repeated blocks). An interesting question is: when do such 1-designs have extensions? Since 2-designs with k = 3 are so common, this problem is too general, and we shall usually impose extra conditions on the extension. Extensions of designs were originally used by Witt, Hughes and Dembowski for studying transitive extensions of permutation groups, and it might be expected that extensions of regular graphs would be useful in the study of doubly transitive groups. This is indeed the case.

If we are extending a design by adding an extra point p, we know all the blocks of the extension containing p - these are obtained simply by adjoining p to all the old blocks - but it is in the blocks not containing p that possible ambiguities arise. Let us then suppose, as a first possibility, that we have a set Δ of triples, called blocks, in which the blocks not containing a point p are uniquely determined in a natural way by those containing p; in particular let us suppose that, when we know which of {pqr}, {pqs} and {prs} are blocks, we can decide whether {qrs} is a block.