Edges in Fibonacci cubes, Lucas cubes and complements

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 $\Gamma\_n$