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I've seen this question several times, and this one caught my attention. I'm now aware that there is no closed formula for this. My knowledge of graph theory is limited, and I wasn't able to find an explanation as to how to use it. So if someone could direct me to a good explanation, or explain it to me that would be great.

And also, I don't understand why the following wouldn't work to calculate the number of graphs:

For a graph with N vertices, we have M = (N * (N - 1))/2 edges we can potentially use. My reasoning was that if K >= N, then we are guaranteed a connected graph. Then for a graph where K >= N, surely there would just be M choose K potential graphs?

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  • $K\geq N$ does not imply the graph is connected. What is true is that if a graph on $N$ vertices is connected, then $K\geq N-1$. – Casteels Aug 30 '15 at 02:33

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"My reasoning was that if K >= N, then we are guaranteed a connected graph." -- This is not true; consider the graph $K_{n/2} + K_{n/2}$, which is a disjoint union of two cliques. This graph has about $n^2/4$ edges, but is clearly not connected.

Gregory J. Puleo
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