Determinants of modular Collatz graphs and variants

TL;DR

This work provides explicit determinant formulas for the adjacency matrices of modular Collatz graphs and Conway's amusical permutation graphs by linking the determinants to multiplicative orders modulo divisors of the modulus. A key tool is a permutation-pair framework that reduces determinant calculation to cycle data of associated permutations, yielding zero determinants when even-order cycles occur and powers of two otherwise, with signs dictated by modulus residues. The results extend to general PNQ-type functions and reveal deep connections between graph structure, cycle decompositions, and number-theoretic orders, while also establishing strong connectedness across modular cases. Although the formulas illuminate why determinants behave erratically as the modulus varies, they do not resolve the underlying Collatz or amusical conjectures themselves, but provide a precise arithmetic explanation for these modular graphs' spectral properties.

Abstract

The determinants of modular Collatz graphs and the modular Conway amusical permutation graph are determined, and some interesting number theoretic properties are described.

Figures (9)

• Figure 1: Examples of modular Collatz graphs for $N = 2, 3, \ldots, 10$. Blue edges come from the rule $n \to \dfrac{n}{2}$, green edges come from the rule $n \to \dfrac{3n+1}{2}$.
• Figure 2: Adjacency matrix for $N = 71$, where the two bands ${\color{blue}\epsilon_2}, {\color{OliveGreen}\epsilon_3}$ are represented with blue and green dots respectively (the positions where a $1$ is in the matrix), and a red dot where they intersect (the position where a $2$ is in the matrix).
• Figure 3: Classes for eigenvalues $\pm 1$.
• Figure 4: $\pm \log_2(\pm\det C_N)$ for odd integers $N$ with $3 \leq N < 10,000,000$. Blue dots indicate prime $N$, red dots indicate composite $N$. The $\pm$ indicates the sign of $\det C_N$.
• Figure 5: The development of the $\pi_3$-cycles $\!\!\!\pmod{5}$ (left) into acyclic components $\!\!\!\pmod{3 \cdot 5}$ (middle) and $\!\!\!\pmod{3^2 \cdot 5}$ (right).

Theorems & Definitions (39)

• Definition 1
• Definition 2
• Definition 3
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• Definition 4
• proof