Brendan McKay has already done the work for finding all non-isomorphic graphs of n variables that can be found here (under Simple Graphs): http://cs.anu.edu.au/~bdm/data/graphs.html

I believe this was done using polya enumeration, which I understand the basics of. I would like to expand on this, and allow self loops in these graphs. So, i'd like to find all non-ismorphic graphs of n variables, including self loops. This will be directly used for another part of my code and provide a massive optimization. I'm just not quite sure how to go about it.

To be clear, Brendan Mckay's files give all non ismorphic graphs, ie in edge notation,

1-2 1-3

is a graph with an edge between vertex 1 and 2, and 1 and 3. I want this list to also include self loops, ie:

1-2 1-3 1-1

or

1-2 1-3 1-1 2-2

I want the minimum number of graphs, so all non-ismorphic ones. How can I go about finding them, hopefully using the data Brendan McKay has available for simple graphs?