L1 distortion lower bound for an expander graph

Let a connected $d$-regular expander graph have $n$ vertices, edge-expansion constant $h>0$, and $d\ge 2$. For $n\ge 16d^2$, its least bi-Lipschitz distortion into L1 space satisfies the explicit boundNormalize an embedding so that each image distance is at least the graph distance and at most $D$ times it. Every image edge has length at most $D$. The L1 Poincare inequality for an expander graph then gives $D\ge (h/d)\overline{d}$, where $\overline{d}$ is the mean ordered-pair graph distance.

A ball of radius $r$ has at most $1+d+\cdots +d^r\le 2d^r$ vertices. Take $r=\lfloor \log (n/4)/\log d\rfloor$. Its size is at most $n/2$, so at least half the vertices are farther than $r$ from each fixed vertex. Hence $\overline{d}\ge r/2$. The size assumption implies $r\ge (\log n)/(2\log d)$, proving the bound. For a fixed-degree family with expansion bounded below, distortion therefore grows at least proportionally to $\log n$.