Asymptotic Properties of Fibonacci Cubes and Lucas Cubes

It is proved that the asymptotic average eccentricity and the asymptotic average degree of both 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 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/sqrt{5})/ rm 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 average ratio of the numbers of Fibonacci strings with a 0 (a 1, respectively) in a given position, where the average is taken over all positions, converges to ({varphi^{2}}) , and likewise for Lucas strings.