# Obtaining all connected simply laced graphs with Sage for GAP

I work with GAP and have not much experience with Sage. I wonder whether there is an easy way to obtain a program with Sage that gives as output a list of all directed simply laced acyclic connected graphs on n points up to equivalence that is readable for GAP (via the GAP-package QPA). I know that Sage can generate the list of such combinatorial collections in an easy way but I do not know how to present it in the needed output and how to obtain all restrictions on the graphs. With acyclic I mean acyclic as an directed graph (but in case this makes problems, you can also assume acyclic as undirected graphs).

Here a more detailed description of what the program should do:

Input: A natural number n >= 2.

Output: The list of directed simply laced acyclic connected graphs on n points up to equivalence. Here we call two such graphs equivalent in case they have the same underlying undirected graph (in case it is a problem to do it up to equivalence, we can first look at the problem without up to equivalence). It is not important what the concrete orientation of the arrows of a representative in an equivalence class is.

Note that graphs are also often called quivers. GAP reads those graphs as `Quiver(n, [[p1, p2, "a1"], ...,[pm, pn, "ar"]])`

where we name the points `pi`

and the arrows between them as `aj`

for some index sets `i`

and `j`

. (I hope it will be clear with the following examples).
Here how to output should look in the first 2 cases for n <= 3 so that it is readable for GAP:

n=2: `[Quiver(2,[[1,2,"a1"]])];`

n=3: `[Quiver(3,[[1,2,"a1"],[2,3,"a2"]]),Quiver(3,[[1,2,"a1"],[2,3,"a2"],[1,3,"a3"]])];`

(note for example here here that the quiver `Quiver(3,[[1,2,"a1"],[2,3,"a2"]])`

is equivalent to the quiver `Quiver(3,[[1,2,"a1"],[3,2,"a2"]])`

since they are equal when looking at their undirected graphs)

So for n=2 there is one equivalence class, while for n=3 there are two (so that for n=3 the output is a list of two elements readable by GAP).