Discrete Extremal Length
Suppose that is some graph and is a collection of paths in . There are two variants of extremal length in this setting. To define the edge extremal length, originally introduced by R. J. Duffin, consider a function . The -length of a path is defined as the sum of over all edges in the path, counted with multiplicity. The "area" is defined as . The extremal length of is then defined as before. If is interpreted as a resistor network, where each edge has unit resistance, then the effective resistance between two sets of veritces is precisely the edge extremal length of the collection of paths with one endpoint in one set and the other endpoint in the other set. Thus, discrete extremal length is useful for estimates in discrete potential theory.
Another notion of discrete extremal length that is appropriate in other contexts is vertex extremal length, where, the area is, and the length of a path is the sum of over the vertices visited by the path, with multiplicity.
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