I've been given the following problem as homework:
Q: How many graphs are there (up to isomorphism) with score (3, 3, 3, 3, 3, 3, 6)? Also, do this for (3, 3, 3, 3, 3, 3).
The "score" is the ordered list of degrees for the vertices in a graph; that is, the vertex degree sequence. For example, if a graph has score (3, 3, 3, 3, 3, 3), this means it has 6 vertices, each with degree 3.
I'm honestly pretty lost as to how to start here. "Up to isomorphism" means ignoring isomorphisms of the same graph (I'm guessing).
My attempts so far: I realize that for the first score, one of the points (with degree 6) is connected to all others, and they're all arranged in some way.
What approach do I take to finding all graphs with some score up to isomorphism?
Note: Please do NOT give me the solution to the question since this is homework. Just looking for tips and strategies.