Finding isomorphism classes of graphs, given $|V|, |E|$, degree sequence, etc. In this particular question I'm asked to find all the isomorphism classes of simple graphs, without loops whose degree sequence is: $3,3,2,2,2$, and to prove the ones I found are all the ones that exist.
I don't know how to do this, and I want to learn how to solve these kind of questions. Also, if I'm given the amount of vertices and edges, is there a closed form for the amount of isomorphism classes?
 A: This is probably not quite the answer you were looking for, but by using some of the gtools included with nauty and Traces, you can just compute the graphs using brute force. By running this code:
geng 5 6 -d2 -D3 | showg -o1

it appears that there are two such graphs:
Graph 1, order 5.
  1 : 4 5;
  2 : 4 5;
  3 : 4 5;
  4 : 1 2 3;
  5 : 1 2 3;

Graph 2, order 5.
  1 : 3 4 5;
  2 : 4 5;
  3 : 1 5;
  4 : 1 2;
  5 : 1 2 3;

A: Just try to draw them. Consider the case where your two degree three vertices are adjacent. How can you attach the others? Now consider the case where the two degree three vertices are nonadjacent. How can you attach the others? (I get a unique graph in each situation)
For a proof, you can use this type of reasoning: consider when the degree 3 vertices are nonadjacent.  Each needs to be connected to 3 other vertices, and there are exactly 3 degree 2 vertices. So we have the bipartite graph $K_{2,3}$.
There is in general no closed formula for the number of nonisomorphic graphs with a given degree sequence, it's just trial and error; maybe you will have to use a computer program tp generate them all, then test for which ones are isomorphic.
