How many non-isomorphic graphs with n vertices and m edges are there?

Could someone tell me how to find the number of all non-isomorphic graphs with $m$ vertices and $n$ edges. (The graph is simple, undirected graph)

In my particular problem, $m =20, n=180$

Attempt at solution:

1. Find the total possible number of edges (so that every vertex is connected to every other one) $k = n(n-1)/2 = 20\cdot19/2 = 190$

2. Find the number of all possible graphs: $s = C(n,k) = C(190, 180) = 13278694407181203$

Now, I'm stuck because a huge portion of the above number represents isomorphic graphs, and I have no idea how to find all those that are non-isomorphic...

Solution 1:

First off, let me say that you can find the answer to this question in Sage using the nauty generator. If you're going to be a serious graph theory student, Sage could be very helpful.

count = 0
for g in graphs.nauty_geng("20 180:180"):
    count = count+1
print count

The answer is 4613. But, this isn't easy to see without a computer program.

At this point, perhaps it would be good to start by thinking in terms of of the number of connected graphs with at most 10 edges. Then, all the graphs you are looking for will be unions of these. You should be able to figure out these smaller cases. If any are too hard for you, these are more likely to be in some table somewhere, so you can look them up.

Connected graphs of order n and k edges is:

n = 1, k = 0: 1
n = 2, k = 1: 1
n = 3, k = 2: 1
n = 3, k = 3: 1
n = 4, k = 3: 2
n = 4, k = 4: 2
n = 4, k = 5: 1
n = 4, k = 6: 1
n = 5, k = 4: 3
n = 5, k = 5: 5
n = 5, k = 6: 5
n = 5, k = 7: 4
n = 5, k = 8: 2
n = 5, k = 9: 1
n = 5, k = 10: 1
.
.
.
n = 10, k = 9: 106
n = 10, k = 10: 657
n = 11, k = 10: 235

I used Sage for the last 3, I admit. But, I do know that the Atlas of Graphs contains all of these except for the last one, on P7.