I'd love your help with this question.
Let $n\geq3$ be a fixed integer. How many non-isomorphic graphs with $V$ vertices and $E$ edges are there where $V+E=n$?
Thank you very much.
This type of enumeration can be achieved using Polya Theory to attain a generating function. The best reference is Riordan, http://books.google.com/books?id=zWgIPlds29UC&lpg=PP1&pg=PP1#v=onepage&q&f=false, page 129. See page 143 for the application to counting graphs by |E| and |V|.
If would like to construct such graphs , there is a C program "nauty" developed by Brendan D. McKay, based on his math article http://cs.anu.edu.au/~bdm/nauty/pgi.pdf.