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

The paper deals with some generalizations of Fibonacci and Lucas sequences, arising from powers of paths and cycles, respectively. In the first part of the work we provide a formula for the number of edges of the Hasse diagram of the independent sets of the h power of a path ordered by inclusion. For h = 1 such a diagram is called a Fibonacci cube, and for h > 1 we obtain a generalization of the Fibonacci cube. Consequently, we derive a generalized notion of Fibonacci sequence, called h-Fibonacci sequence. Then, we show that the number of edges of a generalized Fibonacci cube is obtained by convolution of an h-Fibonacci sequence with itself. 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 h 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.