be defined in a similar fashion. A precise definition is given in Chung's survey. Area and volume are measured by set sizes. For every subset A of the graph G one defines as the set of vertices in with a neighbor in A. A d-dimensional isoperimetric inequality is now defined by (This MathOverflow question provides more details.) The graph analogs of all the examples above hold but the definition is slightly different in order to avoid that the isoperimetric dimension of any finite graph is 0: In the above formula the volume of is replaced by (see Chung's survey, section 7). The isoperimetric dimension of a d-dimensional grid is d. In general, the isoperimetric dimension is preserved by quasi isometries, both by quasi-isometries between manifolds, between graphs, and even by quasi isometries carrying manifolds to graphs, with the respective definitions. In rough terms, this means that a graph "mimicking" a given manifold (as the grid mimics the Euclidean space) would have the same isoperimetric dimension as the manifold. An infinite complete binary tree has isoperimetric dimension ∞. Consequences of isoperimetry A simple integration over r (or sum in the case of graphs) shows that a d-dimensional isoperimetric inequality implies a d-dimensional volume growth, namely where B(x,r) denotes the ball of radius r around the point x in the Riemannian distance or in the graph distance. In general, the opposite is not true, i.e. even uniformly exponential volume growth does not imply any kind of isoperimetric inequality. A simple example can be had by taking the graph Z (i.e. all the integers with edges between n and n + 1) and connecting to the vertex n a complete binary tree of height |n|. Both properties (exponential growth and 0 isoperimetric dimension) are easy to verify. An interesting exception is the case of groups. It turns out that a group with polynomial growth of order d has isoperimetric dimension d. This holds both for the case of Lie groups and for the Cayley graph of a finitely generated group. A theorem of Varopoulos connects the isoperimetric dimension of a graph to the rate of escape of random walk on the graph. The result states Varopoulos' theorem: If G is a graph satisfying a d-dimensional isoperimetric inequality then where is the probability that a random walk on G starting from x will be in y after n steps, and C is some constant.