Resistance values under transformations in regular triangular grids

TL;DR

This paper investigates how resistance values in triangular grid graphs change during circuit transformations, providing a conjecture and proven case about resistance patterns after multiple reductions.

Contribution

It extends prior work by analyzing non-asymptotic resistance behavior during grid reductions and proves a key case of the main conjecture.

Findings

Identifies resistance patterns in diagonals after multiple reductions

Proves a special case of the main conjecture

Improves notation and proof techniques from previous work

Abstract

In [Evans, Francis 2022; Hendel] the authors investigated resistance distance in triangular grid graphs and observed several types of asymptotic behavior. This paper extends their work by studying the initial, non-asymptotic, behavior found when equivalent circuit transformations are performed, reducing the rows in the triangular grid graph one row at a time. The main conjecture characterizes, after reducing an arbitrary number of times an initial triangular grid all of whose edge resistances are identically one, when edge resistance values are less than, equal to, or greater than one. A special case of the conjecture is proven. The main theorem identifies patterns of repeating edge resistances arising in diagonals of a triangular grid reduced $s$ times provided the original grid has at least $4s$ rows of triangles. This paper also improves upon the notation, concepts, and proof techniques introduced by the authors previously.