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On the Average Resistance of n-circuits

Mehdi Nikopour Deilami, Bohdan Zhelyabovskyy

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.

On the Average Resistance of n-circuits

Abstract

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

Paper Structure

This paper contains 8 sections, 19 theorems, 63 equations, 2 figures, 5 tables.

Table of Contents

  1. Quantifying Circuits
  2. Inverses
  3. Properties of $C_i(n)$
  4. The Generating Function of $Q_n$
  5. Resistance: Circuits & Biscuits
  6. Final Remarks
  7. Generalization
  8. More $C_i(n)$ Properties

Key Result

Theorem 1.1

The number of circuits generated by a partition $p$, $N(p)$ can be recursively counted by: where counts the number of $n$-circuits $\forall n\in\mathbb{N}$.

Figures (2)

  • Figure 1: $\mathcal{O}_4$ — the $4$-omnicircuit.
  • Figure 2: The first few values of $M_n$ connected by Bézier curves. The dashed line is $y = 1$.

Theorems & Definitions (41)

  • Theorem 1.1: Partition-based $n$-circuit Generation
  • proof
  • Theorem 2.1: Circuit-Multiplicative Inversion
  • proof
  • Corollary 2.1.1: The Lower Bound
  • proof
  • Theorem 3.1: Omnicircuit Meta-Counting Property
  • proof
  • Corollary 3.1.1
  • proof
  • ...and 31 more