I am trying to make a list of all subgraphs up to isomorphism of $K_n$. (Not of all the $2^{\tbinom{n}{2}}$ subgraphs; only up to isomorphism.) I have two questions:
Is there a formula which will tell me how many subgraphs are there for a given n? If no general formula exists are there specific answers for $n<20$, and what are they.
If I want to make a list of subgraphs which avoids the complement of any subgraph (of $K_n$ up to isomorphism) already in the list, and is maximal possible, how do I do that? Can someone suggest an algorithmic way to do this. I plan to write a computer program to generate this list.
Thanks.