On Thursday, 27 December 2012 20:59:45 UTC+1, Benjamin Jones wrote:
>
> On Thu, Dec 27, 2012 at 11:39 AM, Jernej Azarija
> <[email protected]<javascript:>>
> 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. 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.
> I guess what you want is the automorphism group along with it's action
> on the set of vertices of the graph.
>
>
> 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?
>
> to get the abstract group back along with the set of orbits. Also, by
> setting `translation=True` you can also get a dictionary back that
> provides translation from vertices {0, 1, ..., n} to the domain set of
> the permutation group (a subset of {1, ... , n+1}).
>
> --
> Benjamin Jones
>
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