Resistance bounds for finite vertex-transitive graphs

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $G$ be a finite connected simple vertex-transitive $d$-regular graph, with $N=|V(G)|\ge2$ and diameter $\operatorname{diam}(G)$ for unit edges. Give every edge electrical resistance $1$. The effective resistance $R_G(u,v)$ is the voltage drop between $u,v$ when unit current enters at $u$ and exits at $v$ (and $R_G(u,u)=0$).

Statement recorded in the supplied source.

For each degree $d$, is there a constant $C_d<\infty$ depending only on $d$ such that \[R_G(u,v)<C_d+\frac{\operatorname{diam}(G)^2\log N}{N}\quad(u,v\in V(G))?\] In addition, for every sequence of finite connected vertex-transitive graphs $G_n$ with $\operatorname{diam}(G_n)=o(|V(G_n)|)$, prove \[\max_{u,v\in V(G_n)}R_{G_n}(u,v)=o(\operatorname{diam}(G_n)).\] [S2]

Short English statement

Bound all pairwise resistances in finite transitive graphs by degree and diameter, and show resistance is negligible relative to sublinear diameter.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10000059 (AMR-099-0059). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Itai Benjamini, Euclidean vs. Graph Metric, version dated 16 July 2012; original author manuscript preserved in Arquivo.pt capture of 31 December 2020, Section 5, Conjecture 5.2 and following sentence, p. 5. https://arquivo.pt/noFrame/replay/20201231041538id_/http://www.wisdom.weizmann.ac.il/~itai/erd100.pdf.