Doesn't quite work

Sadly, using the degree sequence as a kind of 'mask' on canonically numbered matrices isn't enough. I should have guessed, it was too simple to be true :(

The smiles for all matrices in the (2, 2, 2, 2, 2, 3, 3) partition are:

C1CC2CC(C1)C2

C1CC2CC(C1)C2

C1C2CC12.C1CC1

C1CC2CCC1C2

C1CC2CCC1C2

C1CC1CC2CC2

C1CCC2CC2(C1)

C1CCC2CC2(C1)

and (ignoring the third one, which is disconnected) there are only 3 or 4 different simple graphs in the list.

Perhaps I would have known all this if I knew what automorphisms were, or orbits of elements in sets, or any of that...