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A graph is termed amenable if it has zero edge isoperimetric constant. (Sometimes the vertex isoperimetric constant is used instead; if it has bounded degree, this doesn't matter).
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A group is termed amenable if it has an amenable Cayley graph. There are many equivalent definitions for this; here's one: Group G is amenable iff it has an invariant mean.
Amenability is an important characterisation of graphs and groups. It says that the rate of growth of the object is subexponential.