L1 Poincare inequality for an expander graph ($\sum a_{xy}\Vert f(x)-f(y){\Vert }_1\ge \frac{h}{|V|}{\sum }_{x,y}\Vert f(x)-f(y){\Vert }_1$)

If $G$ has adjacency matrix $A=(a_{xy})$ and expansion $h$, then every $L^1$-valued map on its $n$ vertices satisfiesFor scalar functions this follows from the layer-cake formula applied above and below a median; integration over the $L^1$ coordinate proves the vector-valued form.

For a nonnegative measurable function $g$,On a graph, applying this identity to $|g(x)-g(y)|$ expresses its total variation as the integral of the edge boundaries of the superlevel sets.

If a $d$-regular graph on $n$ vertices has expansion $h$, its $L^1$ distortion obeysThe Poincare inequality controls average image distance by edge lengths, while bounded-degree ball growth makes the average graph distance comparable to $\log n/\log d$.