Have you ever wanted to test some hypothesis or constructions involving (weakly) connected directed graphs? In that case, it can be quite resource consuming to generate canonically labelled representatives as you go, for instance, in sage. Furthermore, there is a good chance that you are going to need the representatives again and again. After having spent my fair share of computing power to generate these representatives, I went hunting for an online database where I could just download them — but I did not find any. So I created such a database myself using sage, possibly saving you around 12 hours of computations (see: github.com/immanuel-albrecht/databases).

The SQL query

SELECT`arcs`

, COUNT() AS) AS`nbr_of_digraphs`

FROM (SELECT`ID`

, COUNT(`arcs`

FROM`weakly_connected`

AS tmp1 GROUP BY`ID`

) AS tmp2 GROUP BY`arcs`

yields:

arcs nbr_of_digraphs 1 1 2 4 3 12 4 53 5 237 6 1,306 7 7,537 8 47,913 9 322,253 10 2,297,874

Now, let’s compare this to the corresponding OEIS-entry:

1, 4, 12, 53, 237, 1306, 7537, 47913, 322253, 2297874, ...

So it appears that the canonical labeling implemented in sage is working as expected 🙂