Is there another way to define this kind of graph?

Question

Let there be $3m$ (where $m$ is any counting numbers and $m\ge{2}$) copies of $C_4$. we denote each copy of $C_4$ as $C_4(i),\quad 1\le i \le 3m $. Let $v_j(i)\in V(C_4(i)),\quad 1\le j \le 4$ be adjacent to a vertex $v_r(k)\in V(C_4(k)),\quad 1\le k\le 3m,\ 1\le r \le 4$ with $i\neq k$ and $r$ and $j$ may not be necessarily equal and no vertex other in $C_4(i)$ is adjacent to another vertex of $C_4(k)$. We denote this graph as $\Gamma_m$. Here is an example where $m=3$:

Is there another way to define this kind of graph?

thanks!