Isoperimetric Inequalities for Infinite Regular Graphs

Question

Say we have the two-dimensional regular graph $G=(\mathbb{Z}^2,S)$ with generator $S$ such that for every vertex we have $\operatorname{deg}(v)=\lvert S\rvert=\text{const.}$ for all $v\in G$. If we pick a subgraph $A\subseteq G$ and look at the edge expansion $\partial A$ (number of edges leaving the subset), we can show that there are suitable functions $\Phi$ such that the following inequality holds: $$\lvert \partial A\rvert\geq\Phi(\lvert A\rvert).$$ I know that for continuous systems such isoperimetric inequalities cannot be reversed, but for discrete systems (and in a sense graphs are one) I could see an inequality of the type $$\lvert \partial A\rvert\leq\Psi(\lvert A\rvert)$$ existing. Are there results convering this issue?

Answer 1

(more of an extended comment than a proper answer) What kinds of results specifically are you looking for?

It's also not 100% clear to me what you mean by $(\mathbb{Z}^2, S)$. Is this a cayley graph of $(\mathbb{Z}^2, +)$ with generating set $S$?

In general, in an $r$-regular graph, you'll have $|\partial A| \leq r|A|$, and that bound is sharp. If you want $A$ to be connected, you can get $|\partial A| \leq r|A| - |A|+1$ ($A$ must induce a spanning tree).

More generally, suppose $A$ is $k$-connected. Then it must induce a subgraph of minimum degree $k$, which thus has at least $\frac{k|A|}{2}$ edges, so $|\partial A| \leq r|A| - \frac{k|A|}{2}$.