**Definition:** $\kappa(G)$ is defined as the minimum
number of vertices whose removal from $G $ results in a disconnected graph.A graph $G$ is said to be $k$-connected
if $\kappa(G) ≥ k$.
<br><br>
**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 symmetric matrix of the graph created by  vertices of $(G-x)$ which are  adjacent to $x$ and $D$ is the symmetric matrix of the graph created by  vertices of $(G-x)$ which are not adjacent to $x$. $C,D$ are sub-graphs of graph $G$,$|V(C)|>|V(D)|$ where $|V(C)|,|V(D)|$ are total  vertices number  of graph  $C,D$ respectively.

$$ A_x 
=
\left( \begin{array}{ccc}
C & E & 0 \\
E^{T} & D & 1\\
0 & 1 & 0\\
\end{array} \right) 
$$
 again,  this process can be done recursively, where $A=C$. This recursive process can be done maximum    $log_2(|V(G)|)$ times. 
<br>
**Question:**  *In this recursive process, 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 ?*
 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