Consider an $n \times n$ grid graph $G$. Define the following quantity
\begin{equation} m(G) = \text{max}\bigg\{\frac{|E|_{H'}}{|V|_{H'}},~ H' \subseteq G, ~~|V|_{H'} > 0 \bigg\}, \end{equation}
where $|V|_{H'}$ and $|E|_{H'}$ are the number of vertices and edges of the graph $H'$ respectively. I am trying to compute this for the grid graph.
Here's my approach. If I take the entire grid, then I have $n^2$ vertices and $2n^2 -2n$ edges. So, $m(G)$ is at least $2 - \frac{2}{n}$. Is this more or less tight or can we do something better?