Isoperimetric Dimension of Graphs
The isoperimetric dimension of graphs can 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. The isoperimetric dimension of any finite graph is 0. 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 ∞.
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“God cannot be seen: he is too bright for sight; nor grasped: he is too pure for touch; nor measured: for he is beyond all sense, infinite, measureless, his dimension known to himself alone.”
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