Expander Graph from Hypergraph

I came up with this problem while thinking about an optimizing compiler.

Let $$H$$ be a hypergraph. From this we construct a graph $$G_H$$ as follows the vertices are the hyperedges of the hypergraph. There is an edge in $$G_H$$ whenever the hyperedges $$e_1$$ and $$e_2$$ have a vertex in $$G$$ in common.

I would like to know what I can say about the Cheeger constant of $$G_H$$.

In fact I have a family $$H_n$$, so this will go into whether there is a nonzero lower bound that works for all the $$G_{H_n}$$.