vertex isoperimetric constant (thing)

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.