I use the Sage automorphism_group function to find the group that leaves an adjacency matric, say C, invariant under a similarity transformation. But sometimes it seems the results are for matrices on a vertex space that is not the same as the original, but is reordered.
A simple example on a 3-vertex graph: o--o--o.
with index order 1 0 2
Adjacency matrix= C =
[0 1 1]
[1 0 0]
[1 0 0]
The group is the identity and a reflection about the center vertex:
[1 0 0] [0 1 0]
[0 1 0] and [1 0 0] = R
[0 0 1] [0 0 1]
I would have expected the reflection to be,
[1 0 0]
[0 0 1]
[0 1 0],
but R is what Sage gives.
Sure enough, RCR^-1 gives the following,
[0 1 0]
[1 0 1]
[0 1 0]
which is not the original C matrix. However the R transformation leaves the matrix
[0 0 1]
[0 0 1] = C'
[1 1 0]
invariant. The last matrix is the C matrix where the vertices are indexed in reverse order.
Does Sage have some canonical way it orders the vertices in graphs and groups?
A bit more info, maybe. The order of the irreducible representation table seems to give a hint. If the trivial representation is the first row (which it is for some graphs), it appears (from a few tests) that the vertices retain their original order to align with C. If the trivial representation is the last row then the matrix C' is the adjacency matrix (suggesting a reordering of the vertices). I have no idea why this should be related if it is.
Bottom line for me is I cannot apply any matrices from the group that come from the matrix() method to C or other objects in my original vertex space since they are ordered differently. I don't see how to find the vertex order sage uses. Any help or insight appreciated.
