I know there will be 9 vertices of degree 6, which means there are 54 edges. But then how will I figure out the number of non-isomorphic graphs?
They can be generated using
geng which is packaged with Nauty using the command
./geng 9 -d6 -D6. Feeding this into
showg will show the adjacencies of the graphs, which we draw as follows:
We color the non-edges red and identify the complement graphs below each graph. We can readily see why it's easier to count the 2-regular graphs and take their complements.
Since isomorphims map edges to edges, and non-edges to non-edges, two graphs are isomorphic if and only if their complements are isomorphic.