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arXiv · 1602.06002

Sum rules for effective resistances in infinite graphs

Greg Markowsky·José Luis Palacios
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Abstract

Extending work of Foster, Doyle, and others, we show how the Foster Theorems, a family of results concerning effective resistances on finite graphs, can in certain cases be extended to infinite graphs. A family of sum rules is then obtained, which allows one to easily calculate the sum of the resistances over all paths of a given length. The results are illustrated with some of the most common grids in the plane, including the square, triangular, and hexagonal grids.

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Greg Markowsky, José Luis Palacios. 2016-02-18. Sum rules for effective resistances in infinite graphs. https://doi.org/10.1088/1742-5468/aa6503

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