On the Fibonacci $(p,r)$-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