I'm a non-mathematician working with applied graph theoretic tools. I need to figure out how many possible graphs exist for a problem of mine (in a graph structure optimization problem, quickly said).
In essence, my question is: how many non-isomorphic, connected graphs are there on $n$ vertices with $k$ edges?
Where, in my case, $n=50$ and $k=180$.
I had a hard time finding a straight forward answer to this question (other than iterating through a catalog of graphs, where $n$ is usually pretty small). Is there any way to compute such answer?
Thanks!