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Hi!

1) As @Nathann pointed out, you can install nauty by using

$ sage -i nauty

Once nauty is installed you can just iterate through connected graphs on 11 vxs with

sage: for g in graphs.nauty_geng('-c 11'):
          test_property(g)

2) There is a graph database, but apparently it only contains graphs up to 7 nodes (up to 8 in an optional package). Btw, I did not get any errors when running the example, what version of sage are you running?

3) It might not be worth incorporating this data into sage, it is even faster to generate it using geng than to download it or even writing it to the hard drive! :)

Although 1) might solve the problem, my suggestion would be to use geng from the command line and generate the g6 strings for all connected graphs on 11 vertices and store them in a file. Then just use sage to iterate through this file generating one graph at a time and testing properties.