On electric resistances for distance-regular graphs

TL;DR

This paper studies electric potentials on distance-regular graphs, revealing that with high valency, points tend to be nearly equidistant when distance is measured by electric resistance, extending previous findings.

Contribution

It extends prior results by demonstrating that in large valency distance-regular graphs, all points are nearly equidistant under electric resistance measurement.

Findings

Points are nearly equidistant in high valency graphs

Electric resistance defines a metric close to graph distance

Main theorem generalizes previous results

Abstract

We investigate the behavior of electric potentials on distance-regular graphs, and extend some results of a prior paper. Our main result, Theorem 4, shows(together with Corollary 3) that if distance is measured by the electric resistance between points then all points are close to being equidistant on a distance-regular graph with large valency. A number of auxiliary results are also presented.