The *edge isoperimetric constant* of a graph G=(V,E) is a measure of its rate of growth. It is defined by

ι_{e}(G) = inf_{F⊂V finite} |Γ_{e}(F)|/|F|

where Γ

_{e}(F) is the number of

edges {f,g}∈E with f∈F and g∉F.

- The square d-dimensional grid
**Z**^{d}
- ι
_{e}(**Z**^{d})=0: If we take F_{m}={(x_{1},...,x_{d}): |x_{i}|≤m} then there are 2*d*m^{d-1} edges leaving the cube F_{m}, but its volume is m^{d}. So ∀m, ι_{e}(**Z**^{d}) ≤ |Γ_{e}(F_{m})| / |F_{m}| = 2*d*m^{d-1} / m^{d} = 2*d/m -> 0.
- The infinite binary tree B
- ι
_{e}(B)=1: If B_{k} is the first k levels of B, then there are 2^{k-1} vertices at the last level of B_{k}, so 2^{k} vertices leave B_{k}. Since there are B_{k}=2^{k}-1, ι_{e}(B)≤1.
On the other hand, if F is *any* finite subset of the vertices of B, we must show at least |F| edges leave F. Indeed, suppose F is connected and has more than 1 vertex. F itself is a tree; we will show it has at least as many edges leaving it as it has vertices. Every vertex of F has degree 1 or 2 or 3. If it has degree 1, it contributes 2 edges which leave it. If it has degree 2 and is not the root, it contributes one edge which leaves it; this has no effect on the balance, so we can just "elide" it from F, replacing it by an edge, with no effect on the balance of edges leaving F - vertices in F. If it has degree 3 or is the root, it's just one vertex which connects to twice as many subtrees. So everything balances, and |Γ_{e}(F)|≥|F|. If F consists of more than one connected component, we just apply this argument to each component separately.

A graph with zero isoperimetric constant is called *amenable*. Otherwise, it's nonamenable.