I am not a mathematician. I've been working on a problem for some time, and I can't seem to be able to grasp the solution.
I need to find an algorithmic way to solve for the number of connected, labeled graphs (I believe those are the correct terms) that can be drawn, given k edges and n vertices.
Most recently, I've tried to solve for the maximum number of vertices, and work my way down by removing edges.
For example, starting with 5 vertices and 10 edges, there is only one graph that can be drawn. 10 of those edges can be removed to create a unique graph, making the solution for (5,9) 10. For (5,8), the solution I get this way is 45 (9 + 8 + ...), and for (5,7) I get 120, or (sigma(8) + sigma(7) + ... + 1), and so on.
This solution seemed to have worked in my head, but when put to the test it does not pass.