Some new results about Fibonacci p-cubes

The Fibonacci cube ${\mathrm{\Gamma }}_n$ is the subgraph of the hypercube $Q_n$ induced by vertices with no consecutive $1$s. Recently Jianxin Wei and Yujun Yang introduced a one parameter generalization, Fibonacci $p$-cubes ${\mathrm{\Gamma }}_n^p$, which are subgraphs of hypercubes induced by strings where there is at least $p$ consecutive $0$s between two $1$s. In this paper we first prove the expression conjectured by the authors for the cube polynomial of ${\mathrm{\Gamma }}_n^p$. 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 two new ones, the Mostar index $Mo({\mathrm{\Gamma }}_n^p)$ and the Irregularity $\text{irr}({\mathrm{\Gamma }}_n^p)$.