Mike,
However, I haven't been able to figure out a labeling of vertices that actually makes it work. In fact, it looks as though this construction starts with a cyclic 10x10 matrix and then adds two parity bits (which isn't to say it can't be done).
Take a look at this diagram: https://upload.wikimedia.org/wikipedia/commons/7/77/Icosahedron_petrie.png Label the vertices of the decagon from 1 to 10 (in cyclic order), and the remaining vertices 11 and 12 (there are only two ways to do so, and precisely one of them will give the generator matrix in my* illustration). * Sorry about not responding to your comment on my talk page; I wasn't logged into Wikipedia at the time and subsequently forgot about it.
I can label the vertices however I want and get an adjacency matrix for that labeling; is there a way to figure out the permutation given the two matrices?
Yes! https://rjlipton.wordpress.com/2015/11/04/a-big-result-on-graph-isomorphism/ Best wishes, Adam P. Goucher