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Limiting Behavior of Resistances in Triangular Graphs

Russell Jay Hendel

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, $Δ$--Y, and Y--$Δ$ 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.

Limiting Behavior of Resistances in Triangular Graphs

Abstract

Barrett et al studied resistance labels of electrical circuits whose underlying graphs when embedded in the Cartesian plane has the form of an -grid, rows of upright triangles. Proofs in Barrett introduced a row-reduction algorithm which uses series, --Y, and Y-- electric transformations to transform an -grid into an 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 -grid uniformly labeled 1 asymptotically produces triangular grids whose sides are labeled with rational multiples of 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 -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.

Paper Structure

This paper contains 10 sections, 19 theorems, 81 equations, 8 figures, 2 tables.

Table of Contents

  1. Background and Goals
  2. Notations, Conventions, and Algorithms
  3. The Conjectures
  4. A Motivating Example
  5. Needed Functions
  6. The Three Main Theorems
  7. Proof of the Symmetry Theorem
  8. Proof of the Edge-Factor Symmetry Duality Theorem
  9. Proof of Theorem \ref{['the:edgefactors']}(ii), Case $i=c-1.$
  10. Completion of proof of Theorem \ref{['the:edgefactors']}

Key Result

Proposition \oldthetheorem

For integer $c \ge 1$ and $1 \le j \le c-1,$

Figures (8)

  • Figure 1: A 3-grid embedded in the Cartesian Plane constructed using Definition \ref{['def:ngrid']} (Left Panel) with the paper's conventions on row and diagonal coordinates (Right Panel).
  • Figure 2: Definition and orientation conventions for the $\Delta$-Y and Y-$\Delta$ circuit transformations. $S, T, U$ are vertex labels; $L, R, B$ are edge labels standing for left, right, and base edge. The $Y$ legs are labeled with an intuitive clock notation.
  • Figure 3: Illustration of two row reductions, Definition \ref{['def:rr']}, on a 3-grid whose edges are uniformly labeled with 1.
  • Figure 4: A $3$ grid labeled using \ref{['equ:tl11']}-\ref{['equ:edgeratiorelationships']}
  • Figure 5: Two applications of row reduction to an initial non-one 3-grid (Figure \ref{['fig:3rgrid']}) resulting in a non-one 2-grid and 1 grid.
  • ...and 3 more figures

Theorems & Definitions (53)

  • Definition 1
  • Definition 2
  • Example 1
  • Definition 3
  • Example 2
  • Conjecture 1
  • Conjecture 2
  • Conjecture 3
  • Conjecture 4
  • Conjecture 5
  • ...and 43 more