 When formula_4, the symmetries of the rook's graph are formed by independently permuting the rows and columns of the graph, so the automorphism group of the graph has formula_26 elements.When formula_9, the graph has additional symmetries that swap the rows and columns, so the number of automorphisms is formula_28.Any two vertices in a rook's graph are either at distance one or two from each other, according to whether they are adjacent or nonadjacent respectively.