  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 $\varphi^2$.
