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.
