Say you have a graph with 4 vertices, I would know that it can have a maximum of $\begin{pmatrix}4\\2\end{pmatrix} = 6$ edges. What I'm not sure about how to find is how many ways you can connect the vertices if the graph has no loops, and is non-isomorphic.
After manually finding the ways to create these graphs, I observed the following:
| # of edges | # of graphs |
|---|---|
| 0 | 1 |
| 1 | 1 |
| 2 | 2 |
| 3 | 3 |
| 4 | 2 |
| 5 | 1 |
| 6 | 1 |
There must be a way to predict this with permutations/combinations, but I'm just not sure how to change the question into a way that I can calculate it.
