Sum rules for effective resistances in infinite graphs. (3rd April 2017)

• Record Type: Journal Article
• Authors: Markowsky, Greg; Palacios, José Luis
• Is Part Of: Journal of statistical mechanics. (2017:Apr.)
• Publication Date: 2017-04-03
• Subjects: 12; Statistical mechanics -- Periodicals; Mechanics -- Statistical methods -- Periodicals; 530.1305
• DOI: 10.1088/1742-5468/aa6503
• Languages: English
• ISSNs: 1742-5468

Abstract

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.