Some new results about Fibonacci p-cubes
Michel Mollard
TL;DR
This work resolves a conjectured expression for the cube polynomial of Fibonacci $p$-cubes $\Gamma_n^p$, establishing $C_{\Gamma_n^p}(x)=\sum_{a=0}^{\left\lfloor \frac{n+p}{p+1}\right\rfloor} {n-ap+p\choose a}(1+x)^a$ and deriving the corresponding $c_k$ counts. It then introduces and analyzes the distance cube polynomial $D_{\Gamma_n^p}(x,q)$, providing its closed form, generating function, and $c_{k,d}$ counts. The paper further completes the invariant set by deriving explicit expressions for the Wiener index $W(\Gamma_n^p)$ and Mostar index $Mo(\Gamma_n^p)$ in terms of Fibonacci $p$-numbers, and proves a clean formula for irregularity: $\mathrm{irr}(\Gamma_n^p)=2\sum_{d=1}^p |E(\Gamma_{n-d}^p)|$. Collectively, these results place Fibonacci $p$-cubes within the daisy-cube framework and elucidate their isometric subgraph structure, extending known properties of classical Fibonacci cubes to the $p$-parameter family.
Abstract
The Fibonacci cube $Γ_n$ 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 $Γ_n^p$, 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 $Γ_n^p$. 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 two new ones, the Mostar index $\mathit{Mo}(Γ_n^p)$ and the Irregularity $\irr(Γ_n^p)$.