  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
