A Physics Perspective on the Resistance Distance for Graphs

Abstract

The notion of resistance distance as a convenient metric for graphs was introduced in Klein (J Math Chem 12:81–95, 1993). It is inspired by the concept of equivalent resistance for electrical circuits and has numerous applications, in particular, in organic chemistry, physics and random walks on graphs. Besides, computing resistance distance of various circuits has always been of interest for electrical engineers. In this paper, we provide a brief review of the concept and a physics perspective on resistance distance, highlighting some useful analytical methods for computing it. To some extend, these methods generalize and build on top of the results presented in Bapat (Math Stud 68(1–4):87–98, 1999, Indian J Pure Appl Math 41(1):1–13, 2010) and Kagan (Am J Phys 83:53–63, 2015). We then illustrate these methods using graphs with rotational symmetry as an example. The same analysis can be applied to computations of the complex impedance of AC-circuits of the same circular topology and can be used to investigate resonance phenomena therein. At the end, we discuss the concept of resistance distance in the context of the Weisfeiler–Leman stabilization.