A conjecture of Biggs concerning the resistance of a distance-regular graph

TL;DR

This paper proves Biggs' conjecture that the resistance between any two points in a distance-regular graph with valency greater than 2 is bounded by twice the resistance between adjacent points, identifying the sharp constant.

Contribution

It confirms Biggs' conjecture, determines the exact constant for the resistance inequality, and analyzes graphs that nearly violate the conjecture.

Findings

The resistance bound is exactly twice the resistance between adjacent points.

The sharp constant for the inequality is established.

Graphs nearly violating the conjecture are characterized.

Abstract

Previously, Biggs has conjectured that the resistance between any two points on a distance-regular graph of valency greater than 2 is bounded by twice the resistance between adjacent points. We prove this conjecture, give the sharp constant for the inequality, and display the graphs for which the conjecture most nearly fails. Some necessary background material is included, as well as some consequences.