The

*vertex isoperimetric constant* of a

graph G=(V,E) is a measure of its

rate of growth similar to the

edge isoperimetric constant. It is defined by

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

where Γ

_{v}(F) is the number of vertices g∈V for which

∃{f,g}∈E with f∈F.

For a graph with bounded degrees, ι_{v}=0 iff ι_{e}=0. In this case, the graph is amenable, else it's nonamenable.

The d-dimensional grids **Z**^{d} are all amenable; trees (other than lines) are all nonamenable. See edge isoperimetric constant for calculations of that constant, which are easily adapted to yield the vertex isoperimetric constant.