Vertex and edge orbits of Fibonacci and Lucas cubes

TL;DR

This paper analyzes the symmetry orbits of vertices and edges in Fibonacci and Lucas cubes, providing explicit formulas for orbit sizes and counts under automorphism group actions, using dihedral transformations and primitive strings.

Contribution

It determines the orbit structures and sizes for Fibonacci and Lucas cubes, introducing new formulas and methods involving dihedral transformations and primitive strings.

Findings

Set of vertex orbit sizes for Lucas cubes includes divisors of n and 2n.

Number of vertex orbits of size k (odd divisor of n) involves Möbius function and Fibonacci numbers.

Number of edge orbits in Lucas cubes equals the number of vertex orbits in Fibonacci cubes of size n-3.

Abstract

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