A number of strongly regular graphs of several types derived from combinatorial design were identified by Goethals and Seidel (1970).
Theorem 2.4 of Goethals and Seidel (1970) identifies a family of strongly regular graphs corresponding to the existence of a block design with parameters and a Hadamard matrix of order . Some of these graphs are implemented in the Wolfram Language as GraphData["GoethalsSeidelBlockDesign", k, r].
Theorem 2.7 with leads to a strongly regular graph on 105 vertices with parameters which is the second subconstituent of the second subconstituent of the McLaughlin graph. This graph is distance-regular but not distance-transitive with intersection array and graph spectrum . This graph is implemented in the Wolfram Language as GraphData["GoethalsSeidelGraph105"].
Theorem 5.2 identifies a set of five strongly regular graphs having vertex degree equal to vertex count summarized in the following table (where the graph spectrum uses the normal adjacency matrix, not the versions appearing in Goethals and Seidel 1970).
number | name | graph spectrum | regular parameters | |
2 | 253 | |||
3 | 77 | M22 graph | ||
6 | 176 | |||
7 | 56 | Gewirtz graph | ||
9 | 120 |
Some of these graphs are implemented in the Wolfram Language as GraphData["GoethalsSeidelTacticalConfiguration", k] using the numbering scheme above.
Theorem 5.3 identifies the strongly regular graph with vertices of degree 100 now known as the Higman-Sims graph.
Theorem 6.4 identifies a strongly regular graph on 2048 vertices.