In the long term, I think the right solution is to copy what is done for
Galois groups of number fields: depending on a keyword option to
automorphism_group(), we should return either the abstract permutation
group (as is done now) or a group equipped with an action on the edges and
vertices of the graph.
David
On Fri, Dec 28, 2012 at 2:36 AM, Jernej Azarija <azi.std...@gmail.com>wrote:
>
>
> On Thursday, 27 December 2012 23:51:07 UTC+1, Benjamin Jones wrote:
>
>> On Thu, Dec 27, 2012 at 1:07 PM, Jernej Azarija <azi.s...@gmail.com>
>> wrote:
>> >
>> >
>> > On Thursday, 27 December 2012 20:59:45 UTC+1, Benjamin Jones wrote:
>> >>
>> >> On Thu, Dec 27, 2012 at 11:39 AM, Jernej Azarija <azi.s...@gmail.com>
>> >> wrote:
>> >> > Hello!
>> >> >
>> >> > I apologize for posting this question here but somehow I am not
>> allowed
>> >> > to
>> >> > drop questions to sage-support. Moreover I do not feel confident
>> enough
>> >> > to
>> >> > post this thing as a bug on the trac wiki.
>> >> >
>> >> > Working with a large graph G on ~250 vertices I have noticed that
>> >> > elements
>> >> > of the automorphism group of G permute ~50 vertices and that most
>> >> > vertices
>> >> > are fixed by any automorphism. Hence most orbits of the automorphism
>> >> > group
>> >> > contain just singletons. However sage simply discards all vertices
>> that
>> >> > are
>> >> > fixed by the automorphism . In my case this resulted in an
>> "incomplete"
>> >> > orbit containing just 50 elements. An extreme case happens when one
>> >> > deals
>> >> > with an asymmetric graph
>> >> >
>> >> > ===
>> >> > sage: G = graphs.RandomRegular(7,50)
>> >> > sage: G.vertices()
>> >> > [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
>> 19,
>> >> > 20,
>> >> > 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37,
>> 38,
>> >> > 39,
>> >> > 40, 41, 42, 43, 44, 45, 46, 47, 48, 49]
>> >> > sage: G.automorphism_group().domain(**)
>> >> > {1}
>> >> > sage: G.automorphism_group().orbits(**)
>> >> > [[1]]
>> >> > ===
>> >> >
>> >> > This of course is not the desired result since one assumes orbits
>> >> > partition
>> >> > the group.
>> >> >
>> >> > Is this a bug or am I simply missing some parameter to resolve this
>> >> > issue?
>> >> >
>> >>
>> >> I don't think this is a bug.
>> >
>> >
>> > Hello! Thanks for your reply!
>> >>
>> >>
>> >> It looks to me like G.automorphism_group() is returning an abstract
>> >> permutation group. For a lot of random graphs this is going to be the
>> >> trivial group "Permutation Group with generators [()]" (a random graph
>> >> is likely to have no symmetry). The natural (non-empty) domain for the
>> >> action of such a group is a singleton set and there is of course only
>> >> one orbit there. Notice that G.automorphism_group().domain(**)
>> returns
>> >> {1}, it's the domain of a permutation group on {1, ... , n}.
>> >
>> > I am not sure this is consistent with the mathematical definition of
>> the
>> > domain of a group acting on a set S.
>>
>> G.automorphism_group() is not returning "a group acting on a set S",
>> merely a permutation group. Observe that the trivial group is
>> isomorphic to the trivial permutation subgroup of {1} as well as {0,
>> 1, ... , 50}.
>>
>> > Even *if* I take this convention for
>> > granted, it becomes a mess if I try to obtain the orbits of a
>> > vertex-stabilizer. Being more concrete:
>> >
>> > sage: G = graphs.RandomRegular(7,50)
>> > sage: G.automorphism_group().**stabilizer(1).orbits()
>> > [[1]]
>> >
>> > which is clearly not the desired output.
>>
>> This in consistent with what I said above. In this case
>> G.automorphism_group() is the trivial group (permutation group with no
>> generators). So, G.automorphism_group().**stabilizer(1) is again the
>> trivial group.
>>
>> >>
>> >> One simple thing you can do is call:
>> >>
>> >> sage: A = G.automorphism_group(orbits=**True)
>> >
>> > Yes. Is there a way to extend this answer to the case when I wish to
>> obtain
>> > the orbit of a specific subgroup of the automorphism group?
>> >
>>
>> The documentation (G.automorphism_group?) describes how to get the
>> subgroup of the automorphism group that preserves a given partition of
>> the vertex set.
>
> The documentation about the partition thing is quite shallow (just one
> sentence) and the expected usage does not seem to work:
> ===
> sage: G = graphs.PetersenGraph()
> sage: G.automorphism_group(partition=[[1]])
> ---------------------------------------------------------------------------
> KeyError Traceback (most recent call last)
>
> /home/foo/<ipython console> in <module>()
>
> /home/foo/sage-5.5.rc0/local/lib/python2.7/site-packages/sage/graphs/generic_graph.pyc
> in automorphism_group(self, partition, translation, verbosity, edge_labels,
> order, return_group, orbits)
> 16466 HB = H._backend
> 16467 for u,v in self.edge_iterator(labels=False):
> > 16468 u = G_to[u]; v = G_to[v]
> 16469 HB.add_edge(u,v,None,self._directed)
> 16470 GC = HB._cg
>
> KeyError: 0
>
> =====
>
> does anyone happen to know how is this thing used? I need to compute the
> automorphism group that fixes the specified vertex (the stabilizer of a
> vertex v of the automorphism of G)
>
> Other than that it would depend on what subgroup you
>> want. Check out the generic group methods that construct subgroups.
>>
>> Also, see this discussion:
>> https://groups.google.com/**forum/?fromgroups=#!topic/**
>> sage-support/HX0QfXgwO5s<https://groups.google.com/forum/?fromgroups=#!topic/sage-support/HX0QfXgwO5s>
>>
>> --
>> Benjamin Jones
>>
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