Limiting behavior of resistances in triangular graphs

Certain electric circuit can be perceived as undirected graphs whose edges are 1-ohm resistances. Ohm's law allows calculation of equivalent single resistances between two arbitrary points on the electric circuit. For graphs embeddable in the plane, there are four functions that allow the implementation of Ohm's law and calculation of equivalent resistances. Consequently, no knowledge of electrical engineering is needed for this talk. It is a talk about interesting properties of graphs whose edges have specific resistances and which allow reduction to other graphs. Interesting results are possible when the underlying graph belongs to certain families. For example the resistance between two corners (degree-two vertices) of a graph on $n$ edges consisting of $n − 2$ triangles arranged in a line is ${n-1\over5}+{4\over5}{F_{n-1}\over L_{n-1}}$ with $F$ and $L$ representing the Fibonacci and Lucas numbers respectively. This presentation explores triangular graphs of $n$ rows of equilateral triangles. These triangular graphs were mentioned in passing in one paper with a conjecture on the equivalent resistance between two corners. In this presentation we present new computation methods, allowing reviewing more data. It turns out that the limiting behavior of these $n$-row triangular grids (as $n$ goes to infinity) has unexpected simply described behavior: The sides of individual triangles are conjectured to asymptotically equal products of basically fractional linear transformations and $e^{-1}$. We also introduce new proof methods based on a simple verification method.

Russell Jay Hendel