Vertex and edge orbits of Fibonacci and Lucas cubes

The Fibonacci cube $Γ_n$ is obtained from the $n$-cube $Q_n$ 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 $Λ_n$ 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 $Λ_n$ is $\{k \ge 1;\ k \divides n\} \cup\, \{k \ge 18;\ k \divides 2n\}$, the number of the vertex orbits of $Λ_n$ of size $k$, where $k$ is odd and divides $n$, is equal to $\sum_{d\divides k}μ\left(\frac{k}{d}\right) F_{\lfloor \frac{d}{2}\rfloor + 2}$, and the number of the edge orbits of $Λ_n$ is equal to the number of the vertex orbits of $Γ_{n-3}$. Dihedral transformations of strings and primitive strings are essential tools to prove these results.

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