For a strongly regular graph, there are exactly 3 eigenvalues, all nonzero (I believe). One has multiplicity 1, which means the other two have pretty high multiplicities. There are tables that give these eigenvalues and multiplicities:
For example, the Schlaefli graph is order 27 but has an eigenvalue of order 20.
My question is, are there other known graphs (families, types, or just single graphs) that have large multiplicities of eigenvalues? When I check a random graph in Sage, it seems the max multiplicity is mostly 1.