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A symmetry of a graph X is a permutation of the vertices that also happens to be a permutation of the induced edges. In particular, the distances between vertices are preserved by a symmetry. Show that the set of symmetries of X is a permutation group of V(X). Compute the cycle index of the group for the Petersen graph.
Please help. How do you show that something is a permutation group? I have never calculated the cycle index for a graph before and we don't have any examples like this in our notes. How do I do it for a graph?