The term "Goethals-Seidel graphs" is used in this work for strongly regular graphs of several types derived from combinatorial designs and 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.
