On the Fibonacci $(p,r)$-cubes

TL;DR

This paper introduces a new topological structure called Fibonacci $(p,r)$-cube, compares it with the existing one, and explores their properties, including recursive structure, distance invariants, and degree bounds.

Contribution

It establishes that the Fibonacci $(p,r)$-cube is a distinct structure from the original, and analyzes its topological properties and relationships with the original cube.

Findings

Fibonacci $(p,r)$-cube is a new topological structure different from the original.

Both structures exhibit recursive properties and are partial cubes and median graphs.

Distance invariants and degree bounds are characterized for these cubes.

Abstract

In this paper, first it is shown that the "FSibonacci $(p,r)$-cube"(denoted as $I\Gamma_{n}^{(p,r)}$) studied in many papers, such as \cite{OZY}, \cite{K1}, \cite{OZ}, \cite{KR} and \cite{JZ}, is a new topological structure different from the original one (denoted as $O\Gamma_{n}^{(p,r)}$) presented by Egiazarian and Astola $\cite{EA}$. Then some topological properties of $I\Gamma_{n}^{(p,r)}$ and $O\Gamma_{n}^{(p,r)}$ are studied, including the recursive structure of them, the cubes $O\Gamma_{n}^{(p,r)}$ which are partial cubes and median graphs, some distance invariants of $I\Gamma_{n}^{(p,r)}$ and $O\Gamma_{n}^{(p,r)}$, and the maximum and minimum degree of these two types of cubes. Finally, several problems and conjectures on $I\Gamma_{n}^{(p,r)}$ and $O\Gamma_{n}^{(p,r)}$ are listed