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automorphism group of graphs

asked 2013-12-20 01:12:04 -0500

AG gravatar image

When I create a graph in SAGE and then look at its automorphism group, the permutation group obtained acts on points $1,\ldots, n$ (where $n$ is the number of vertices of the graph), but it is not clear to me what the correspondence is between the vertices of the graph and these points. For example the code and output

sage: X=Graph(); X.add_edges([(0,1),(0,2)]); 
sage: X.automorphism_group().list()
[(), (1,2)]

suggests that the vertices 0, 1 and 2 correspond to points 3, 1 and 2, respectively. Similarly,

sage: X=Graph(); X.add_edges([(0,1),(1,2)]); 
sage: X.automorphism_group().list()
[(), (2,3)]

also suggests that vertex name 1 corresponds to point 1. The example

sage: X=Graph(); X.add_edges([('a','b'),('b','c')]);                      
sage: X.automorphism_group().list()
[(), (2,3)]

suggests that vertex name 'b' corresponds to the point 1, and 'a' and 'c' correspond to 2 and 3, respectively.

I was wondering in general what the correspondence is between the vertex labels and the positive integer points that the automorphism group of the graph acts on, especially when the vertex names are not integers but arbitrary (letters or permutations), as in the latter example given above. This correspondence can be determined for small graphs by inspection, but I was wondering what the correspondence is in general.

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answered 2013-12-20 01:36:54 -0500

fidbc gravatar image

What version of sage are you using? It seems that in sage 5.13 that is no longer an issue.

sage: g=graphs.CubeGraph(2)
sage: g.automorphism_group()
Permutation Group with generators [('01','10'), ('00','01')('10','11')]

According to the current code in the automorphism_group method, it seems that the labelling is according to the order given by the vertices method (provided you do not specify a partition as argument).

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I'm using version 5.3

AG gravatar imageAG ( 2013-12-20 01:57:21 -0500 )edit

It was solved 7 months ago. Really you should upgrade your version of Sage, many nice things have been included/fixed since ! ;-)

Nathann gravatar imageNathann ( 2013-12-20 04:26:33 -0500 )edit

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Asked: 2013-12-20 01:12:04 -0500

Seen: 417 times

Last updated: Dec 20 '13