Asymptotic Properties of Fibonacci Cubes and Lucas Cubes

Abstract

It is proved that the asymptotic average eccentricity and the asymptotic average degree of both Fibonacci cubes and Lucas cubes are (5 + √ 5)/10 and (5 − √ 5)/5, respectively. A new labeling of the leaves of Fibonacci trees is introduced and it is proved that the eccentricity of a vertex of a given Fibonacci cube is equal to the depth of the associated leaf in the corresponding Fibonacci tree. Hypercube density is also introduced and studied. The hypercube density of both Fibonacci cubes and Lucas cubes is shown to be (1 − 1/ √ 5)/ log 2 ϕ, where ϕ is the golden ratio, and the Cartesian product of graphs is used to construct families of graphs with a fixed, non-zero hypercube density. It is also proved that the average ratio of the numbers of Fibonacci strings with a 0 resp. a 1 in a given position, where the average is taken over all positions, converges to ϕ 2 , and likewise for Lucas strings.

On domination-type invariants of Fibonacci cubes and hypercubes

“…Afterwards they have been extensively studied and found additional applications, see the survey [23]. The interest for Fibonacci cubes continues, recent research of them includes asymptotic properties [24], connectivity issues [7], the structure of their disjoint induced hypercubes [14,30], the (non)-existence of perfect codes [5], and the q-cube enumerator polynomial [31]. From the algorithmic point of view, Ramras [29] investigated congestion-free routing of linear permutations on Fibonacci cubes, while Vesel [34] designed a linear time recognition algorithm for this class of graphs.…”

The structure of $k$-Lucas cubes

“…Basic graph theoretic properties of Λ n appear in [13]. The average degree of a vertex in Γ n and Λ n are computed in [10] and the induced d-dimensional hypercubes Q d in Γ n and Λ n are studied in [3,9,11,[14][15][16].…”

Upper Bounds on the Domination and Total Domination Number of Fibonacci Cubes

“…Their usage in theoretical chemistry and some results on the structure of Fibonacci cubes, including representations, recursive construction, hamiltonicity, the nature of the degree sequence and some enumeration results are presented in [3]. Characterization of induced hypercubes in Γ n are considered in [4][5][6][7][8] and many additional new properties of Fibonacci cubes are given in the literature, see for example [9][10][11]. Furthermore, the domination number (see, Section 2) of Γ n is first considered in [12,13].…”