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
ιv(G) = infF⊂V finitev(F)|/|F|
where Γv(F) is the number of vertices g∈V for which {f,g}∈E with f∈F.

For a graph with bounded degrees, ιv=0 iff ιe=0. In this case, the graph is amenable, else it's nonamenable.

The d-dimensional grids Zd 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.