So I have been painting polyhedra these days. I paint their vertices with different colors and then I paint the edges with colors that depend on the colors of the vertices in either end, and it turns out there are a lot of ways to do that. I can paint everyone the same color, but that is boring. Using two colors already produces a vast richness of isomers.
Each possible polyhedron of, say, nine vertices, will have different isomers for which the colors are the same, but the specific vertices and edges that receive them are different. What characterizes and isomer is that the colors have to be the same and the fact that isomers can't be converted into one another by a swap of vertices that corresponds to rigid rotations in space.
Each isomer has a special adjacency matrix (SAM) that is constructed based on their color scheme. Since the colors are the same, they are just swapped around, and the colors of the edges follow along this vertex swap, the SAM of each isomer is a permutation of the SAM $A_0$ of a particular reference permutation.
I think this means that, given some SAM $A$ we can write that $P A P^{-1} = A$ where matrix $P$ is a permutation matrix. Is that so, though? Is it $P A P^{-1}$ or just $P A$?
Whatever the case I would then very much like to know how can I compute this $P$ matrix given $A$ and $A_0$.
It feels like it should be a trivial computation, but rummaging through the questions already asked I found a lot of more advanced inquiries, but nothing so straightforward. I wondered whether the thing is entirely impossible or so trivial that nobody had needed to ask it here.
