Optimal Resistor Networks

TL;DR

This paper investigates the minimal average resistance in resistor networks modeled by graphs, revealing new constructions that outperform traditional extremal graphs like stars and regular graphs, especially near average degree 3.

Contribution

It introduces novel constructions for resistor networks that achieve lower average resistance, and establishes the asymptotic equivalence between rooted and unrooted graph resistance minimization.

Findings

New resistor network configurations outperform classical extremal graphs.

Asymptotic equivalence between rooted and unrooted graph resistance problems.

Improved bounds on average resistance for graphs with average degree near 3.

Abstract

Given a graph on n vertices with m edges, each of unit resistance, how small can the average resistance between pairs of vertices be? There are two very plausible extremal constructions -- graphs like a star, and graphs which are close to regular -- with the transition between them occuring when the average degree is 3. However, one of our main aims in this paper is to show that there are significantly better constructions for a range of average degree including average degree near 3. A key idea is to link this question to a analogous question about rooted graphs -- namely `which rooted graph minimises the average resistance to the root?'. The rooted case is much simpler to analyse than the unrooted, and one of the main results of this paper is that the two cases are asymptotically equivalent.