11 - Classification of rank 3 graphs

Summary

The classification of rank 3 graphs is due to Foulser-Kallaher-Kantor-Liebler-Liebeck- Saxl and others. We describe that result in this chapter. We provide the relevant group-theoretic theorems (without proof) and identify all graphs that appear with graphs that were mentioned (and in most cases explicitly constructed) in previous chapters. This way, we give all pairs (Γ,), with Γ a strongly regular graph and 𝐺 a group of automorphisms of Γ acting rank 3. We also provide a table of the parameters of the sporadic rank 3 graphs, which have up to 531441 vertices, and a similar table for all rank 3 graphs up to 1024 vertices. Finally, we tabulate small primitive rank 4–10 strongly regular graphs which do not have Latin square parameters.