There is quite a good discussion of the problem in the first chapter of the text, GRAPHS: An Introductory Approach by Wilson & Watkins. (Because of your location, you can find it in most mathematics libraries.) It turns out that the problem is different for labeled graphs than for unlabeled graphs. You seem to be using labeled graphs, graphs having named vertices . For unlabeled graphs Polya solve the problem in 1935. Wilson & Watkins give what they call “graph cards” at the end of chapter 1: they list all non-isomorphic unlabeled graphs of order 1 to 6. There are 153 non-isomorphic unlabeled graphs of order 6.