I have seen some questions about labeled graphs, but no answers for an unlabeled graph. I calculated for the first four graphs and it seems to be $2n-1$ total graphs for $n$ vertices. I'm not sure if this holds for all possible graphs however. Is there a way to show this?
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I believe it's an extremely open problem, see here. Also, check your formula against the list there :) – pjs36 May 17 '15 at 21:38
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The following MSE link might prove interesting reading. – Marko Riedel May 17 '15 at 21:41
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I'm not sure how you got to your 2n-1 result, as there are 11 graphs on 4 vertices! Also note that there are different numbers for connected and disconnected. A000088 is disconnected, and A001349 is connected. – gilleain May 17 '15 at 22:25