I have been trying to count the number of graphs up to isomorphism which are:

- Simple
- Connected
- Have $n$ edges.

I apologize in advance if there is ample documentation on this question; however, I have found none. Thus far, my best overestimate is: $$g(n) = \sum_{i=x}^y t(i) \cdot \binom{a(i)} { n - i - 1}$$

where:

$g(n) := $ the number of such graphs with $n$ edges,

$t(i) :=$ the number of trees up to isomorphism on $i$ vertices,

$a(i) :=$ the number of non-adjacent vertices in a tree on $i$ vertices.

I have conjectured that: $$a(i) = \sum_{k-1}^i (i - k), \qquad y = n+1,\quad\text{and}$$ $x \geq $ the number of vertices in the complete graph with the closest number of edges to $n$, rounded down.

I have also read that the number of trees including isomorphism with $i$ vertices is $i^{i-2}$, and have placed that as the upper bound for $t(i)$.

And that [according to Wikipedia] there is an estimate for the number of such trees up to isomorphism: $t(i)\sim C \alpha^i i^{-5/2}$ with $C=0.534949606...$ and $\alpha=2.99557658565...$.

What I would like to know is:

A. Is there an answer already found for this question?

B. Is there any information off the top of your head which might assist me?

C. Is this problem incredibly hard?

Again, I apologize if this is not appropriate for this site. I am a sophomore undergraduate student, and I have been trying to answer or estimate this question for use as an upper bound for another larger question that I am working on.

Thanks for the help.