Asymptotic Properties of Fibonacci Cubes and Lucas Cube

TL;DR

This paper investigates the asymptotic properties of Fibonacci and Lucas cubes, including average eccentricity, degree, and hypercube density, revealing connections to Fibonacci words and the golden ratio.

Contribution

It introduces new labeling techniques, studies hypercube density, and establishes asymptotic properties and limits related to Fibonacci and Lucas cubes.

Findings

Asymptotic average eccentricity is (5+√5)/10.

Hypercube density is (1-1/√5)/log₂φ.

Limit normed sum of Fibonacci and Lucas words ratios is φ².

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 $\varphi^2$.