It is well discussed in many graph theory texts that it is somewhat hard to distinguish non-isomorphic graphs with large order. But as to the construction of all the non-isomorphic graphs of any given order not as much is said. So, it follows logically to look for an algorithm or method that finds all these graphs.
A Google search shows that a paper by P. O. de Wet gives a simple construction that yields approximately $\sqrt{T_n}$ non-isomorphic graphs of order n.
( $\{T_n}$ being the number of labeled graphs of order n.)
So, I have the followings to ponder over:
(1) Are there such algorithms or has there been an improvement on the aforementioned algorithm?
(2) Where can I find a collection of non-isomorphic graphs of a given order?
If you allow me, I would also like to extend my question to connected graphs.
Many thanks.
(I am a beginner in Graph theory, so please give answers in not-very-specialized terms.)
Best Answer
The nauty software contains the "geng" program, which enumerates all nonisomorphic graphs of a given order, or only connected ones, or selected on a wide range of other criteria. The method is tuned for practical speed rather than simplicity or theoretical bounds. The author Brendan McKay also has a page where you can download nonisomorphic (connected) graphs up to 10 vertices.