Number of non-isomorphic connected graphs with m edges.

Question

This might be an easy one, but I cant figure it out.

Given $m$ edges how many connected (non-isomorphic) graphs can be drawn. Ofcourse there are no loops, multiple edges etc.

I tried the recurrence method but introduction of cycles complicate it. Also for a given value of $m$, I might calculate the number of such graphs using sage, but I needed an expression.

Am I missing some obvious counting technique?

I doubt there is a single nice formula for this, see oeis:A002905. — dtldarek, Aug 28 '13 at 07:17
Here is a MSE Post that shows how to calculate these values and presents some context and background material. — Marko Riedel, Aug 28 '13 at 19:42

score 1 · Accepted Answer · edited Jun 12 '20 at 10:38

1

There is no exact formula for this. The closest you can get is to use a program like nauty to compute the number of graphs for specific values of the number of vertices and edges.

Example. Computing the number of non-isomorphic connected graphs with $9$ vertices and $13$ edges.

foo@darkstar:~$ geng -c 12 13:13 -u

28908 graphs generated in 0.11 sec

edited Jun 12 '20 at 10:38

Community

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answered Aug 28 '13 at 19:17

Jernej

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Number of non-isomorphic connected graphs with m edges.

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