Limiting Behavior of Resistances in Triangular Graphs

TL;DR

This paper investigates the asymptotic behavior of resistances in triangular grid graphs using a row-reduction algorithm, revealing rational limiting ratios and symmetries, and introduces a mechanical verification method for resistance identities.

Contribution

It provides numerical evidence and proves that the limiting edge-ratios in triangular grids exhibit specific symmetries and algebraic relationships, advancing understanding of electrical resistance patterns in these graphs.

Findings

Repeated row-reduction leads to rational limiting edge-ratios.

Limiting ratios exhibit vertical and rotational symmetries.

A mechanical verification method for resistance identities is introduced.

Abstract

Barrett et al studied resistance labels of electrical circuits whose underlying graphs when embedded in the Cartesian plane has the form of an $n$-grid, $n$ rows of upright triangles. Proofs in Barrett introduced a row-reduction algorithm which uses series, $\Delta$--Y, and Y--$\Delta$ electric transformations to transform an $n$-grid into an $n-1$ grid with equivalent resistances between specified nodes. This paper explores this row-reduction algorithm computationally. The introductory part of the paper presents several conjectures supported by numerical evidence, showing that repeated application of the row-reduction algorithm to an initial $n$-grid uniformly labeled 1 asymptotically produces triangular grids whose sides are labeled with rational multiples of $\frac{1}{e};$ moreover, the ratio of specified consecutive edges in the row-reduced grids are asymptotically described by four rational functions. The main part of this paper studies a family of graphs whose edge labels are determined using these limiting edge-ratios functions arising in the conjectures. The main result proven is that these $n$-grids and their repeated reductions under the row-reduction algorithm possess vertical and rotational symmetries and satisfy the relationships captured by the four edge-ratio functions. Thus, the limiting edge-ratio relationships are local algebraic relationships mirroring the global vertical and rotational symmetries possessed by the underlying graph. Additionally, because row-reduction is local (in contrast to the combinatoric Laplacian which is global) the paper is able to introduce a mechanical verification method of proof for assertions about effective resistance identities.