Some new results about Fibonacci p-cubes

TL;DR

This paper explores properties of Fibonacci p-cubes, a generalization of Fibonacci cubes, providing formulas for their cube polynomial, distance cube polynomial, and new invariants like Mostar index and irregularity.

Contribution

It proves a conjectured formula for the cube polynomial of Fibonacci p-cubes and introduces the distance cube polynomial and new invariants, expanding understanding of these graphs.

Findings

Derived the cube polynomial expression for Fibonacci p-cubes

Determined the distance cube polynomial for Fibonacci p-cubes

Calculated new invariants: Mostar index and irregularity for Fibonacci p-cubes

Abstract

The Fibonacci cube $\Gamma_n$ 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 $\Gamma_n^p$, 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 $\Gamma_n^p$. 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 $\mathit{Mo}(\Gamma_n^p)$ and the Irregularity $\irr(\Gamma_n^p)$.