Notation: $H$ is the adjacency matrix of graph $H'$ respectively. $H_k$ is the block or sub-matrix of matrix $H$. The adjacency matrix of graph $H_k \cup H_e$ (subgraphs of $H'$) is $M_{(k,e)}$ where $M_{(k,e)} =\left( \begin{array}{ccc} H_e & R_{k,e} \\ R_{k,e}^{T} & H_k\\ \end{array} \right) $, where, $R_{k,e}$ is the non symmetric sub-matrix of adjacency matrix $H$. Here, $R_{k,e}$ represents edges between $H_k, H_e$. The matrix $H$ looks like- $$H = \begin{bmatrix} H_{(x)} & R_{(x, x-1)} & R_{(x,x-2)} & \dots & \dots & R_{(x,1)} \\ R_{(x,x-1)^{T}} & H_{(x-1)} & R_{(x-1,x-2)} & \dots & \dots & R_{(x-1,1)} \\ \vdots & \vdots & \vdots & \ddots & \ddots & \vdots \\ R_{(x,1)^{T}} & R_{(x-1,1)^{T}} & R_{(x-2,1)^{T}} & \dots & \dots &H_{1} \end{bmatrix}$$.
Facts: Each $M_{k,e}$ can have exactly $6$ vertices, $3$ vertices in $H_k$ and $3$ vertices in $H_e$. So, $R_{k,e}$ is a non-symmetric matrix of dimension $3 \times 3$ .
It is clear that each distinct $R_{k,e}$ can appear maximum $b$ times where $b \leq n^{9}$, since there are maximum $n^{3}$ different possible $H_k$ and $n^{3}$ different possible $H_e$ .
Problem: I want to prove that-
Each distinct $R_{k,e}$ can appear maximum $\sqrt b \leq n^{3}$ times.
I am assuming the statement is correct.
This is a wrong statement. $ \sqrt b \geq n^{3}$ whatever way you try.