Some new results about Fibonacci p-cubes

Abstract

The Fibonacci cube $\hat{I}{“}_n$ 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 $\hat{I}{“}_n^p$, 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 $\hat{I}{“}_n^p$. 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 $Mo(\hat{I}{“}_n^p)$ and the Irregularity $\irr(Γ_n^p)$.

arXiv admin note: text overlap with arXiv:2202.09068