Context: Given a adjacency matrix A of a $r$ regular graph $G$ . $G$ is $k$ connected.

The matrix A can be divided into 4 sub matrices based on adjacency of vertex $x \in G$. $A_x$ is the adjacency matrix of the graph $(G-x)$, where $C$ is the adjacency matrix of the graph created by vertices which are not adjacent to $x$ and $D$ is the adjacency matrix of the graph created by vertices which are adjacent to $x$. $C,D$ are sub-graphs(regular) of graph $G$,$|V(C)|>|V(D)|$ where $|V(C)|,|V(D)|$ are total vertices number of graph $C,D$ respectively.

$$ A_x = \left( \begin{array}{ccc} C & E \\ E^{T} & D \\ \\ \end{array} \right) $$ again, this process can be done recursively, where $A=D$. This recursive process can be done maximum $log_2(|V(G)|)$ times.
Question: In this recursive process(assuming $C,D$ are always regular in this recursive process),

1)Is it possible to have an $E$ matrix as a zero matrix, i.e. , is it possible to have disconnected sub-graphs $C,D$ under the given condition that $G$ is $k$ connected ?

2)If possible , then what is the maximum number of happening this incident(e.g $|V(G)|/2 $ times maximum)?

This question is connected to this [ post][1].

Request: Please, inform if anything is unclear/undefined . Also, inform , if the question is miss-tagged.Thanks. [1]: http://math.stackexchange.com/questions/1240637/counting-problem-of-combinations-of-symmetric-matrix