2014/07/18 by Али Реза Ашрафи, Jernej Azarija, Ashrafi, Ali Reza +7 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Advanced Mathematical Theories and Applications #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications
paper · pdf · doi:10.48550/arxiv.1407.4962
openalex publication_date 2014/07/18 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28
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 \divides n\ ∪ \k ≥ 18; k \divides 2n\, the number of the vertex orbits of Λn of size k, where k is odd and divides n, is equal to ∑d\divides kμ((k)/(d)) F\lfloor (d)/(2)\rfloor + 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.