Theory of resistor networks: The two-point resistance

Details Theory of resistor networks: The two-point resistance Theory of resistor networks: The two-point resistance

2004-02-14

arxiv.org/abs/math-ph/0402038v2

Journal of Physics A 37, 6653-6673 (2004)

Physics

Mathematical Physics

30 pages, 5 figures now included; typos in Example 1 corrected

Scientific paper

10.1088/0305-4470/37/26/004

The resistance between arbitrary two nodes in a resistor network is obtained in terms of the eigenvalues and eigenfunctions of the Laplacian matrix associated with the network. Explicit formulas for two-point resistances are deduced for regular lattices in one, two, and three dimensions under various boundary conditions including that of a Moebius strip and a Klein bottle. The emphasis is on lattices of finite sizes. We also deduce summation and product identities which can be used to analyze large-size expansions of two-and-higher dimensional lattices.

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