**Upload:** the general question has been answered in the negative. Now I am proposing a more specific question, which is actually the one that I am interested in.

I think that this is a quite natural question, but I was able neither to find anything in literature, nor to find an argument by myself.

Let $G=(V,E)$ be an infinite, locally finite, connected graph (maybe without loops and multiple edges) equipped with the shortest path metric. Given a finite subset $A$ of $V$, let $\partial A$ denote the set of vertex at distance $1$ from $A$. There are many definitions of isoperimetric constant. I am adopting the following one: $\iota(G)$ is the infimum of $\frac{|\partial A|}{|A|}$, when $A$ runs over the finite non-empty subsets of $V$.

Now, take an increasing and covering sequence $A_n$ of finite non-empty subsets of $V$ and define the number $j$ to be the infimum of the $\frac{|\partial A_n|}{|A_n|}$'s.

The general question concerned possible relations between $j$ and $\iota(G)$. In particular, I was interesting in the following

Original question:Is it possible that $\iota(G)=0$ and $j>0$?

It has trivially negative answer: take a tree with degree $\geq3$ and attach a copy of $\mathbb Z$ to one vertex.

The question that I am interested in is actually more specific. Fix a point $x$ and consider the sequence $A_0=x$, $A_1=\partial A_0$, ... $A_n=\partial A_{n-1}$. Define $j(x)$ to be the infimum of the $\frac{|\partial A_n|}{|A_{n-1}|}$'s. Let $j$ be the infimum over $x$ of the $j(x)$'s. What about the following question?

New question 1:Is it possible that there are vertex $x$ and $y$ such that $j(x)\neq j(y)$?

New question 2:Is it possible that $j=0$ and there exists $x$ such that $j(x)>0$?

Basically I am looking for some local version of the isopetric constant...