On Saturday, 5 May 2012 09:51:55 UTC, Nathann Cohen wrote:
>
> Hellooooooooo everybody !!!
>
> Being lost in a foreign country, my hobbies get weirder and weirder. These
> days I find myself playing with graph drawings because of a colleague of
> mine, Thomas Connor.
>
> We are currently trying to plot his graphs in some meaningful way, and his
> graphs are veeeerry symmetrical. Hence we would like to find a way to plot
> them using the information contained in their automorphism group, using
> some properties we think of along the way. Hence, this email : would anyone
> here happen to have examples of nice symmetrical graphs which have nice
> representations ? What we actually need right now (besides good ideas for a
> plotting heuristic) is a collection of some graph to test it on. If you
> have some you like, could you please tell me how to build them through Sage
> ? If we finally get anywhere we will send you back nice pictures to get
> your advice on the drawings produced !
>
For very symmetric (say, vertex and edge-transitive) graphs a meaningful
way to draw would be what is called
variously a collapsed adjacency matrix, a distance distribution diagram,
etc.
That is, you take a meaningful subgroup H, like the stabilizer of a vertex,
of the automorphism group, and basically create a weighted graph, with
multiple arcs, with vertices being orbits of H on vertices, and joining
these new vertices with bi-weighted edges;
e.g. for the Petersen graph such a diagram would look like the following:
[1]-------[3]--------[6]
3 1 2 1 2
where an orbit of length k of H is denoted by [k], and the row of numbers
below the diagram is indicating the (bi-)weights:
as follows, from left to right: there are 3 edges going from [1] to [3],
there is 1 edge going from each element of [3] to [1];
there are 2 edges going from each element of [3] to [6]; there is 1 edge
going from each element of [6] to [3], and finally,
there are 3 edges going from each element of [6] to [6].
In such a way one can "draw" rather huge graphs, see e.g. p.2 of
http://arxiv.org/pdf/math/0012266v1.pdf
Dima
>
> I initially asked Jason who thought some of you may be interested in the
> adventure :-D
>
> Thaaaaaaaaaaaaaaaaaaaaaaaaaaank you for your thoughts ! ;-)
>
> Nathann
>
> P.S. : Oh, and if you already know of some code producing such drawings,
> please share it !
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