# When can the Cheeger constant be well-approximated by Hamming balls''?

Given a graph G, the Cheeger constant is defined by $$\DeclareMathOperator{\Vol}{Vol} h_G := \min_{S \subseteq V, \Vol S \leq (\Vol G)/2} \frac{|\partial S|}{\Vol S}.$$ Here, $\Vol S$ is the sum of the degrees of the vertices in the subgraph induced by $S$. For $v \in V$, let $N_k(v) := \{w \in V : d(v,w) \leq k\}$.

Are there any results that indicate when $h_G$ might be well-approximated by taking $S$ to be some $N_k(v)$, for appropriate choices of $k$ and $v$? The complete graph shows that this cannot always work, but it seems like it might in some cases, such as certain classes of random graphs.

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