So, self loops are not allowed, but parallel edges are allowed. That means that for every (n+1) solution must be a superset of the solution for n edges. I say start with the basic solution of two nodes with all edges connected between them, and disconnect them one at a time to a new node until you are reduced to the last edge. Then iterate for every pair of nodes.
If I want to discover all possible graphs with 10 edges, I'll need to iterate from 2 to 11 nodes, since in the worst case a complete graph with 10 edges will have 11 nodes.
A K11 (complete graph with 11 nodes) will have 55 edges. And I think that a naive algorithm will be impractical in this case. For small problems, OK. But I want to reach at least 12 edges.