Asymptotic Properties of Fibonacci Cubes and Lucas Cube

Abstract: It is proved that the asymptotic average eccentricity and the asymptotic average degree of Fibonacci cubes and Lucas cubes are $(5+\sqrt 5)/10$ and $(5-\sqrt 5)/5$, respectively. A new labeling of the leaves of Fibonacci trees is introduced and 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 $(1-1/\sqrt 5)/\log_2\varphi$, where $\varphi$ 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 limit normed sum of ratios of Fibonacci words and Lucas words with fixed coordinate 0 and 1, respectively, is $\varphi^2$.