The Euler graphs on 11 vertices can be found on Brendan McKay's webpage, the two-graphs on at most 10 vertices are on Edward Spence's webpage.

The 25.112 self-complementary and the 43.849.092 non-self-complementary two-graphs (7zipped graph6 format).

Exactly 15.274 of these are special: they have larger automorphism group than any of the graphs in their switching class.

The 34 self-complementary special and the 7.620 non-self-complementary special two-graphs (7zipped graph6 format).

See the description of the graph6 format.

You may find a 7-zip archiver for Windows or for OS X by following the links.

References: