On the independent subsets of powers of paths and cycles

In the first part of this work we provide a formula for the number of edges of the Hasse diagram of the independent subsets of the hth power of a path ordered by inclusion. For h = 1 such a value is the number of edges of a Fibonacci cube. We show that, in general, the number of edges of the diagram is obtained by convolution of a Fibonacci-like 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 subsets of the hth power of a cycle ordered by inclusion. For h = 1, and n > 1, such a value is the number of edges of a Lucas cube.