An Introductory Survey of Recursions in the Computation of Resistance Distance

Abstract

This paper presents an introduction and expository account of a beautiful, current, and active application of recursions to the computation of resistance distance. Resistance distance, also referred to as effective resistance, is a well-known graph metric that arises naturally by considering a graph as an electrical circuit; heuristically resistance distance measures both the number of paths between two vertices in a graph and the cost of each path. This topic finds applications in a rich array of fields including social, biological, ecological, and transportation networks, chemistry, graph theory, numerical linear algebra, and engineering. A variety of methods are used in the field to determine resistance distance including recursive, mathematical, and graphical techniques. Sequences familiar to the readers of the Fibonacci Quarterly such as the Fibonacci and Lucas sequences appear quite often in results in the literature. Twenty five to forty years ago there were a handful of papers on resistance that appeared in the Fibonacci Quarterly and the Proceedings and recently papers on the subject have appeared again. It is hoped that this introductory expository account will interest readers of the Quarterly to renew interest in this current and active field.