  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 $\Gamma\_n$.
