Variance of Resistance of "Line-Circle-Line" Graphs

TL;DR

This paper analyzes the variance growth of resistance in a specific graph model, deriving its order based on a stochastic growth process with particular conditions on the function involved.

Contribution

It establishes the order of the variance of resistance in 'line-circle-line' graphs using a stochastic growth model with new analytical bounds.

Findings

Variance of resistance grows exponentially with base approximately 2.125.

Derived conditions on the function f influence the variance order.

Provides a mathematical framework for resistance variance in complex graph structures.

Abstract

We find the order of the variance of the growth model $X_{n+1} = X_n+X'_n + f(X''_n,X'''_n)$, where all the variables $X_n,X'_n,X''_n$ and $X'''_n$ are i.i.d., $X_0$ takes the values $1$ and $2$ with equal probability and $f$ is positive, monotone non-decreasing and satisfies conditions which, roughly speaking, pertain to its first and second order partial derivatives. For an appropriate choice of $f$ we obtain that the variance of the effective resistance between the endpoints of the "line-circle-line" graph $G_n$ is of order $\bigl(2 + \frac{1}{8} + {\scriptscriptstyle\mathcal{O}} (1)\bigr)^n$.