# Reference request: maximal Cheeger constant for 3-regular graphs

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 $$n\to+\infty$$?

Meanwhile, here's a simple proof that $$\liminf h_n \le 1$$.
Consider a connected induced subgraph $$H$$ with $$n_1,n_2,n_3$$ vertices of degree 1,2,3, respectively. Since $$H$$ is connected, we have $$n_1+2n_2+3n_3\ge 2(n_1+n_2+n_3)-2$$ (sharp for a tree); that is, $$n_3\ge n_1-2$$, say $$n_3=n_1-2+\delta$$. Also $$|\partial H|\le 2n_1+n_2$$, so in summary, the Cheeger constant is at most $$\frac{2n_1+n_2}{2n_1+n_2-2+\delta},$$ which is bounded above by $$1+o(1)$$ as $$H$$ grows in size.
ADDED: I just realised there is a much simpler way to show that $$h_n\le 1$$ except for tiny $$n$$: Take $$H$$ to be a cycle.