L1 distortion lower bound for an expander graph
= L1 distortion lower bound for an expander graph
{c}
If a $d$-regular graph on $n$ vertices has expansion $h$, its $L^1$ distortion obeys
$$
c_1(G)\geq\frac{h}{2d\log d}\left(\log\frac n2-1\right).
$$
The Poincare inequality controls average image distance by edge lengths, while bounded-degree ball growth makes the average graph distance comparable to $\log n/\log d$.