1. 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).
2. 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.

A*me"na*ble (#), a. [F. amener to lead; (L. ad) = mener to lead, fr. L. minare to drive animals (properly by threatening cries), in LL. to lead; L. minari, to threaten, minae threats. See Menace.]

1. Old Law

Easy to be led; governable, as a woman by her husband.

Jacob.

2.

Liable to be brought to account or punishment; answerable; responsible; accountable; as, amenable to law.

Nor is man too diminutive . . . to be amenable to the divine government. I. Taylor.

3.

Liable to punishment, a charge, a claim, etc.

4.

Willing to yield or submit; responsive; tractable.

Sterling . . . always was amenable enough to counsel. Carlyle.

© Webster 1913.