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

edit

Here is some code that might do the job. Please feel free to test it and let me know if this works.

def directg(g6):
    import subprocess
    sp = subprocess.Popen("directg -o", shell=True,
                          stdin=subprocess.PIPE, stdout=subprocess.PIPE,
                          stderr=subprocess.PIPE, close_fds=True)
    stdout = sp.communicate(input='{0}\n'.format(g6))[0]
    d6_strings = stdout.split()
    for d6 in d6_strings:
        yield DiGraph(d6[1:])

def A_base(n):
    return graphs.PathGraph(n).graph6_string()

def D_base(n):
    g = graphs.PathGraph(n-1)
    g.add_edge(n-3,n-1)
    return g.graph6_string()

def E_base(n):
    g = graphs.PathGraph(n-1)
    g.add_edge(2,n-1)
    return g.graph6_string()

def generate_DynkinQuivers(n,kind='A'):
    if kind not in 'AED':
        raise ValueError("Invalid type of Dynkin quiver, should be one of 'A', 'D' or 'E'.")
    if kind == 'A':
        g6_base = A_base(n)
    elif kind=='D':
        g6_base = D_base(n)
    elif kind=='E':
        g6_base = E_base(n)
    output_template = 'DynkinQuiver("{kind}",{size},[{orientation}])'

    for digraph in directg(g6_base):
        orientation = []
        for i in range(n-2):
            if digraph.has_edge(i,i+1):
                orientation.append('"r"')
            elif digraph.has_edge(i+1,i):
                orientation.append('"l"')
            else:
                raise ValueError("Format conversion error.")
        if kind in 'AD':
            if kind=='A':
                other = n-2
            else:
                other = n-3
            if digraph.has_edge(other,n-1):
                orientation.append('"r"')
            elif digraph.has_edge(n-1,other):
                orientation.append('"l"')
            else:
                raise ValueError("Format conversion error. Last edge of kind {0} quiver.".format(kind))
        if kind =='E':
            if digraph.has_edge(n-1,2):
                orientation.append('"d"')
            elif digraph.has_edge(2,n-1):
                orientation.append('"u"')
            else:
                raise ValueError("Format conversion error. Last edge of kind E quiver.")
        yield output_template.format(kind=kind,size=n,orientation=','.join(orientation))
def DynkinQuivers(n,kind='A'):
    return '[{0}]'.format(','.join(generate_DynkinQuivers(n,kind)))

Sample usage:

DynkinQuivers(3,'A')

produces

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

Which can be saved to a file.

It takes about half a second to go up to n=13

%time
output=DynkinQuivers(13,'E')
CPU time: 0.41 s, Wall time: 0.53 s