Edges in Fibonacci Cubes, Lucas Cubes and Complements

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