Some new results about Fibonacci p-cubes

The Fibonacci cube Γ n is the subgraph of the hypercube Q n induced by vertices with no consecutive 1s. Recently Jianxin Wei and Yujun Yang introduced a one parameter generalization, Fibonacci p -cubes Γ n p , which are subgraphs of hypercubes induced by strings where there is at least p consecutive 0s between two 1s. In this paper we first prove the expression conjectured by the authors for the cube polynomial of Γ n p . By a totally different method we then determine a generalization, the distance cube polynomial. We also complete the invariants investigated in the original paper by three new ones, the Mostar index Mo ( Γ n p ) , the Wiener index W ( Γ n p ) and the Irregularity irr ( Γ n p ) .