Generalized Fibonacci and Lucas cubes arising from powers of paths and cycles

In the second part we consider the case of cycles. We evaluate the number of edges of the Hasse diagram of the independent sets of the hth power of a cycle ordered by inclusion. For h=1 such a diagram is called Lucas cube, and for h>1 we obtain a generalization of the Lucas cube. We derive then a generalized version of the Lucas sequence, called h-Lucas sequence. Finally, we show that the number of edges of a generalized Lucas cube is obtained by an appropriate convolution of an h-Fibonacci sequence with an h-Lucas sequence.