I'd love your help with this question.
Let $n\geq3$ be a fixed integer. How many non-isomorphic graphs with $p$ vertices and $q$ edges are there where $p+q=n$?
Thank you very much.
Using the Combinatorica package in Mathematica, the command NumberOfGraphs$[p,q]$ returns the number of non-isomorphic graphs with $p$ vertices and $q$ edges. If you want to implement this yourself, you may want to proceed here first.
Edit: Indeed it is a standard application of Pólya theory to obtain formulas for the number of nonisomorphic graphs with $p$ vertices and $q$ edges. (Counting the number where the total number of vertices and edges is $n$ can be obtained from this.) The standard book on graph enumeration is "Graph enumeration" by Harary and Palmer. There is a web site with many sequences arising from results discussed in the book.