# Finding all simply laced Dynkin graphs with a given number of vertices up to isomorphism

The simply laced Dynkin graphs look as follows https://upload.wikimedia.org/wikipedi... and are known from many classification results in mathematics, like Lie algebra or path algebras of finite representaiton type.

Now I try to use SAGE to obtain all possible directed Dynkin quivers with a given number of simples and underlying type. So the input is one of the symbol A, D or E and a natural number $n \geq 2$ and the output should be all directed Dynkin quivers of type X with n points up to isomorphism (where X is A , D or E). The output should have the following form in the example when the input is X=A and n=3:

[DynkinQuiver("A",3,["r","l"]),DynkinQuiver("A",3,["l","r"]),DynkinQuiver("A",3,["r","r"])].

So the output is a list with 3 entries, which GAP can understand. An entry of the form DynkinQuiver("A",3,["r","l"]) is easy to understand: First A is the type, 3 the number of vertices and ["r","l"] is the orientation (right and left). See https://folk.ntnu.no/oyvinso/QPA/manu... on page 14 how the input is for type D and E. Here another example for type E: Q:=DynkinQuiver("E",7,["r","l","r","l","r","u"]).

My motivation is to use SAGE to get this list of quiver and then I want to use the output of SAGE to put it into GAP, where I can do some calculation with the path algebras of those quiver. I want to use SAGE since GAP can not handle problems with graphs and especially is not able to give the list up to isomorphism. Is there a quick way SAGE can find all such directed graphs up to isomorphism and can do the output in the given form? I have a conjecture that might give a new homological characerisation of those Dynkin graphs so Im really curious to test it for a large class of those graphs.

I offer a 30 Euro reward for a fast code that can solve the problem in a quick way maybe until n=12 or 13. I can pay via paypal or Am azon gift card (or donate to an organisation of your choice in case the organisation has paypal payment methods).

Is $E_n$ defined for all $n\geq 6$? Or is it just for $n\in{6,7,8}$?

@fidbc It is just defined for n=6,7,8 (or more precisly, it is a Dynkin graph just for those values), but of course it would be no harm to have a generalisation for higher n since one chooses the input number n anyway.

What exactly do we mean by "fast code"? Any memory usage constraints? Is a 1 minute wait reasonable for $n=13$?

Guess the purpose of $E_n$ is having the $n$-th vertex anchored at the third vertex of the path from left to right, correct?

@fidbc yes, and 1 minite is of course fine.