Graph which do not satisfy a pseudo-Poincaré inequality

A counterexample is the subgraph of the $\mathbb{Z}^2$ Cayley graph found by taking squares $S_i$ of side $i$ and arranging them along a (near)diagonal in a chain so that each $S_i$ is adjacent to $S_{i+1}$ along a single edge, and no other edges connect any $S_i$ to any other.

Suppose the given inequality held. For a given $n$, choose $i > 10n$ and let $f$ be the characteristic function of $S_i$. Then the right hand side of the inequality is $\sim C n \cdot 2$. On the other hand, for each of the $\sim n^2$ points within distance $n/2$ of the edge between $S_i$ and $S_{i+1}$ we have that $f_n$ is bounded away from $0$ and $1$, and so the left hand side of the inequality is $\gtrsim n^2$.

Why I do not have a relevant counterexample yet, I decided to write an extended comment on related Poincaré and Sobolev inequalities. The purpose was to place the question in the right context, provide a source that contains many related references and mention a result (inequality (*)) in the positive direction that is strictly related to the inequality in the question.

A lot is known about Poincaré inequalities on Cayley graphs of finitely generated groups of polynomial growth.

Denote by $|B|$ volume (number of points) in a of a ball $B$. Denote also by $f_B$ average of $f$ on $B$. Also $|\nabla f(x)|=\sum_{y\sim x}|f(x)-f(y)|$ where $x\sim y$ mean that $x$ and $y$ are connected by an edge.

Theorem. If $G$ is a finitely generated group of polynomial growth i.e. volume of a ball of radius $n$ in the associated Caley graph is $\approx n^d$ for some positive integer $d$. Then for any ball $B$ of radius $n$ the following Sobolev-Poincare inequality is true: $$ \left(\frac{1}{|B|}\sum_{x\in B}|f(x)-f_B|^q\right)^{1/q}\leq C n \left(\frac{1}{|B|}\sum_{x\in B}|\nabla f(x)|^p\right)^{1/p} $$ where $1\leq p<d$ and $1\leq q\leq dp/(d-p)$. The constant $C$ does not depend on $B$.

In particular taking $p=q=1$ yields $$ \sum_{x\in B}|f(x)-f_B|\leq C n \sum_{x\in B}|\nabla f(x)| \qquad\qquad (*) $$

This is similar to your inequality but the difference is that you take the $\ell^1$ norm on the ball and not on the whole graph. Also the graph cannot be arbitrary. You need some structure in order to prove the Poincaré inequality.

For a proof of the above result and many references see:

P. Hajłasz, P. Koskela, Sobolev met Poincaré. Mem. Amer. Math. Soc. 145 (2000), no. 688.