Year: 2024
Type: article
Abstract: Given a graph on vertices with 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 occurring when the average degree is 3. However, in this paper, we 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 that the unrooted, and the one of the main results of this paper is that the two cases are asymptotically equivalent.
Source: Mathematika
DOI: 10.1112/mtk.12278
Authors: J. Robert Johnson, Mark Walters
Institution: Queen Mary University of London
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,780
OpenAlex ID: W4402311969