# Expansion vs Sparsest cut

let $G=(V,E)$ and $S\subsetneq V$ then expansion of set $S$ is

$$\alpha(S)=\frac{|E(S,\overline{S})|}{\min\{|S|,|\overline{S}|)\}}$$ where $\bar{S}=V\setminus{S}$ and $E(S,\bar{S})$ are edges between S and $\bar{S}$. Expansion of graph G is $$\alpha(G)=\min_{\emptyset\neq S\subsetneq {V}}\alpha(S)$$ Sparsity $\mathcal S_G$ of graph G is $$\mathcal S_G=\min_{\emptyset\neq S\subsetneq {V}}\frac{|E(S,\overline S)|}{|S||\overline S|}$$ It is easy to see, that $$\frac{n}{2} \mathcal{S}_G\leq \alpha(G) \leq (n-1)\mathcal{S}_G$$ where $n=|V|$. Solution for Expansion Problem is to find set $S\subsetneq V$ such that $\alpha(S)=\alpha(G)$ Solution for Uniform Sparsest Cut Problem is to find set $S\subsetneq V$ such that $$\frac{|E(S,\overline S)|}{|S||\overline S|}=\mathcal S_G$$. My question is, is there a graph for which there are different solution for Expansion Problem and for Uniform Sparsest Cut Problem? Thank you for any ideas ..