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

Edges in Fibonacci cubes, Lucas cubes and complements