I am trying to take the automorphism group of a finite graph as a permutation group, and then have that permutation group act on R^{vertices of the graph} by permuting coordinates of points. However, when I convert the group elements to permutation matrices, sage seems to sometimes relabel the vertices of the graph, causing the group to act incorrectly. I noticed the problem with the wheel graph. I ran the following four lines in a sagemath jupyter notebook.
W5 = graphs.WheelGraph(5); W5
(I can't include the output since I can't attach pictures, but the important thing is that the middle vertex, which is fixed by all automorphisms, is labelled 0).
W5.adjacency_matrix()
[0 1 1 1 1]
[1 0 1 0 1]
[1 1 0 1 0]
[1 0 1 0 1]
[1 1 0 1 0]
(The adjacency matrix above is included as evidence that the center vertex is indeed indexed 0.)
aut = W5.automorphism_group(); aut.list()
[(), (2,4), (1,2)(3,4), (1,2,3,4), (1,3), (1,3)(2,4), (1,4,3,2), (1,4)(2,3)]
g=aut.an_element(); g
(1,2,3,4)
g.matrix()
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[1 0 0 0 0]
[0 0 0 0 1]
I suspect that the problem occurs because the vertex 0 is fixed by all automorphisms. Notice that when the group element is converted to a matrix, the indexing seems to change from 0,..,4 to 1,..,5 with the last coordinate fixed instead of the first. I also tested the cycle graph, and here the problem does not occur, even when I choose a group element that fixes vertex 0:
C5 = graphs.CycleGraph(5); C5
(Again, I can't include a picture, but the vertices are numbered 0 to 4)
autcycle = C5.automorphism_group(); autcycle.list()
[(),
(0,4,3,2,1),
(0,3,1,4,2),
(0,2,4,1,3),
(0,1,2,3,4),
(1,4)(2,3),
(0,4)(1,3),
(0,3)(1,2),
(0,2)(3,4),
(0,1)(2,4)]
gcycle = autcycle.random_element(); gcycle
(1,4)(2,3)
gcycle.matrix()
[1 0 0 0 0]
[0 0 0 0 1]
[0 0 0 1 0]
[0 0 1 0 0]
[0 1 0 0 0]
I think this is a bug, but if not, I would love some tips on how to achieve my expected outcome. Thank you!
