Incidence relations defining designs and incidence relations induced by designs can sometimes be expressed in terms of graphs. Such graphs usually have a high degree of regularity reflecting the regularity of the corresponding designs.

Strongly regular graphs

Let N be an incidence matrix of a symmetric (v, k, λ)-design. If N is symmetric with zeros on the diagonal, it serves as an adjacency matrix of a graph Γ of order v. This graph is regular of degree k, and for any distinct vertices x and y of Γ, there are exactly λ vertices which are adjacent to both x and y.

If N is a symmetric incidence matrix of a symmetric (v, k, λ)-design with ones on the diagonal, then N – I serves as an adjacency matrix of a regular graph of order v and degree k – 1. For any distinct vertices x and y of this graph, the number of vertices that are adjacent to both x and y is equal to λ – 2 if x and y are adjacent and is equal to λ otherwise.

The graphs we have just described are special cases of strongly regular graphs.