Given a connected graph $G$, the *Cheeger constant* $h(G)$ (a.k.a. *Cheeger number* or *isoperimetric number*) roughly measures the "bottleneckedness" of $G$. See Wikipedia for the precise definition.

I want to have an approximate value of the maximal Cheeger constant among all *trivalent* (or *$3$-regular*) graphs with a large fixed number of vertices, but couldn't find an answer by googling. The precise problem is:

Question:Let $\mathcal{T}_n$ denote the set of all $3$-regular graphs with $n$ vertices. How does $h_n:=\max_{G\in\mathcal{T}_n}h(G)$ behave asymptotically as $n\to+\infty$?

Any comments are appreciated!