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The Fibonacci cube Γ n is obtained from the n-cube Q n by removing all the vertices that contain two consecutive 1s. It is proved that Γ n admits a perfect code if and only if n ≤ 3.

In this paper, we address the following question: when is a finite p-group G self-similar, i.e. when can G be faithfully represented as a self-similar group of automorphisms of the p-adic tree? We show that, if G is a self-similar finite p-group of rank r, then its order is bounded by a function of p and r. This applies in particular to finite p-groups of a given coclass. In the particular case of groups of maximal class, that is, of coclass 1, we can fully answer the question above: a p-group of maximal class G is self-similar if and only if it contains an elementary abelian maximal subgroup over which G splits. Furthermore, in that case the order of G is at most p p+1 , and this bound is sharp.2010 Mathematics Subject Classification. Primary 20E08.

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 ≥ 1; k | n} ∪ {k ≥ 18; k | 2n}, the number of the vertex orbits of Λ n of size k, where k is odd and divides n, is equal to, and the number of the edge orbits of Λ n is equal to the number of the vertex orbits of Γ n−3 when n ≥ 5. Primitive strings, dihedral transformations and asymmetric strings are essential tools to prove these results.

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