Vertex and Edge Orbits of Fibonacci and Lucas Cubes

The Fibonacci cube Γn is obtained from the n-cube Qn by removing all the vertices that contain two consecutive 1s. If, in addition, the vertices that start and end with 1 are removed, the Lucas cube Λn is obtained. The number of vertex and edge orbits, the sets of the sizes of the orbits, and the number of orbits of each size, are determined for the Fibonacci cubes and the Lucas cubes under the action of the automorphism group. In particular, the set of the sizes of the vertex orbits of Λn is {k ≥ 1; k |n} ∪ {k ≥ 18; k | 2n}, the number of the vertex orbits of Λn of size k, where k is odd and divides n, is equal to ∑ d | k μ ( k d ) Fb d2 c+2 , and the number of the edge orbits of Λn is equal to the number of the vertex orbits of Γn−3. Dihedral transformations of strings and primitive strings are essential tools to prove these results.