On electric resistances for distance-regular graphs

Abstract

We investigate the behavior of electric potentials on distance-regular graphs, and extend some results of a prior paper, Koolen and Markowsky (2010) [15]. Our main result shows that if the distance between points is measured by electric resistance then all points are close to being equidistant on a distance-regular graph with large valency. In particular, we show that the ratio between resistances between pairs of vertices in a distance-regular graph of diameter 3 or more is bounded by 1 + 6/k, where k is the degree of the graph. We indicate further how this bound can be improved to 1+ 4/k in most cases. A number of auxiliary results are also presented, including a discussion of the diameter 2 case as well as applications to random walks. (C) 2012 Elsevier Ltd. All rights reserved.