Edges in Fibonacci cubes, Lucas cubes and complements

TL;DR

This paper investigates the irregularity of Fibonacci and Lucas cubes, providing bijective proofs of known formulas, and introduces a constant-time algorithm for computing edge imbalance, also analyzing the complement of Fibonacci cubes.

Contribution

It offers a new bijective proof of irregularity formulas for Fibonacci and Lucas cubes and presents a constant-time algorithm for edge imbalance calculation.

Findings

Irregularity of Fibonacci and Lucas cubes expressed in terms of edges and vertices.

A bijective proof of irregularity formulas based on incident edge pairs.

A constant-time algorithm for computing edge imbalance.

Abstract

The Fibonacci cube of dimension n, denoted as $\Gamma\_n$, is the subgraph of the hypercube induced by vertices with no consecutive 1's. The irregularity of a graph G is the sum of |d(x)-d(y)| over all edges {x,y} of G. In two recent paper based on the recursive structure of $\Gamma\_n$ it is proved that the irregularity of $\Gamma\_n$ and $\Lambda\_n$ are two times the number of edges of $\Gamma\_{n-1}$ and $2n$ times the number of vertices of $\Gamma\_{n-4}$, respectively. Using an interpretation of the irregularity in terms of couples of incident edges of a special kind (Figure 2) we give a bijective proof of both results. For these two graphs we deduce also a constant time algorithm for computing the imbalance of an edge. In the last section using the same approach we determine the number of edges and the sequence of degrees of the cube complement of $\Gamma\_n$.