On the Average Resistance of n-circuits

TL;DR

This paper investigates the average resistance of series-parallel networks with n resistors, establishing bounds, conjecturing convergence to 1.25, and providing proofs of key results.

Contribution

It provides bounds for the mean resistance of n-circuits, conjectures its convergence to 1.25, and offers complete proofs of significant related results.

Findings

Mean resistance M_n lies between 1 and 4.3954 for all n

Conjecture that M_n converges to 1.25 as n increases

Provides complete proofs of important properties of n-circuits

Abstract

$n$-circuits are series-parallel networks composed of exactly $n$ unit resistors. This paper is focused on evaluating the mean resistance of all $n$-circuits, $M_n$, establishing that it lies between $1$ and $4.3954$ for all $n$. We ultimately conjecture that $M_n$ converges to $1.25$ as $n$ grows, based on computational analysis and other intuitive arguments. Although the number of $n$-circuits has been explored quite thoroughly, this paper also provides complete proofs of some important results.