Given a graph G with adjacency matrix A, the set of automorphisms of G is precisely those permutations which preserve every eigenspace of A.
Suppose we examine the restriction of the permutation-matrix representation of the automorphic group to one such subspace. Should one expect this restriction to be reducible or irreducible?
I would like to ignore in this respect the graph with only self-loops, which has only one eigenspace, and the automorphic group, which is all of S_n is reducible to a trivial, and a (n-1)-degree irreps.