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Automorphism group of weighted graph

I know we can use sage to find the group of automorphisms of a graph $G$:

G.automorphism_group().list()


However, the above way can only be used to the unweighted graph. So for example:

G = matrix([[0,10,0],
[10,0,1],
[0,1,0]])
G1 = Graph(G, weighted = True)
G1.show(edge_labels=True )
G.automorphism_group().list()


The result is:

[(), (0,2)]


However, this result is not correct (correct for unweighted case). This is because $AD\neq DA$, where

$$D = \begin{bmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{bmatrix},$$ which is a permutation matrix and

$$A = \begin{bmatrix} 0 & 10 & 0 \\ 10 & 0 & 1 \\ 0 & 1 & 0\end{bmatrix},$$ which is an adjacency matrix.

Can we use SAGE to find the group of automorphisms of a graph?

Automorphism group of weighted graph

I know we can use sage to find the group of automorphisms of a graph $G$:

G.automorphism_group().list()


However, the above way can only be used to the unweighted graph. So for example:

G = matrix([[0,10,0],
[10,0,1],
[0,1,0]])
G1 = Graph(G, weighted = True)
G1.show(edge_labels=True )
G.automorphism_group().list()


The result is:

[(), (0,2)]


However, this result is not correct (correct for unweighted case). This is because $AD\neq DA$, where

$$D = \begin{bmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{bmatrix},$$ which is a permutation matrix and

$$A = \begin{bmatrix} 0 & 10 & 0 \\ 10 & 0 & 1 \\ 0 & 1 & 0\end{bmatrix},$$ which is an adjacency matrix.

Can we use SAGE to find the group of automorphisms of a graph? graph?