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Spectral Antisymmetry of Twisted Graph Adjacency

Ye Luo, Arindam Roy

TL;DR

This work establishes a graph-theoretic prime counting framework in which circuits and primes are classified by homology classes. Central to the approach is a spectral antisymmetry theorem: the spectra of twisted adjacency matrices at characters related by the canonical character \\theta come in negated pairs, with a unique extremal role played by the trivial and canonical twists; this duality is formalized via the character group and formal L-functions. The authors derive trace formulas linking traces of twisted edge adjacencies to counts of circuits in homology classes, yielding Fourier-transform relations between class counts and spectral data, and they obtain asymptotic formulas for cycle and prime-cycle counts in homology classes, including vanishing phenomena in parity and torsion cases. By connecting Ihara zeta/L-functions to a unitary character framework, the work provides new tools for analyzing cycle distributions on graphs, with potential implications for prime counting in graph-theoretic settings and for understanding spectral behavior of complex unit gain graphs.

Abstract

We address a prime counting problem across the homology classes of a graph, presenting a graph-theoretical Dirichlet-type analogue of the prime number theorem. The main machinery we have developed and employed is a spectral antisymmetry theorem, revealing that the spectra of the twisted graph adjacency matrices have an antisymmetric distribution over the character group of the graph with a special character called the canonical character being an extremum. Additionally, we derive some trace formulas based on the twisted adjacency matrices as part of our analysis.

Spectral Antisymmetry of Twisted Graph Adjacency

TL;DR

This work establishes a graph-theoretic prime counting framework in which circuits and primes are classified by homology classes. Central to the approach is a spectral antisymmetry theorem: the spectra of twisted adjacency matrices at characters related by the canonical character \\theta come in negated pairs, with a unique extremal role played by the trivial and canonical twists; this duality is formalized via the character group and formal L-functions. The authors derive trace formulas linking traces of twisted edge adjacencies to counts of circuits in homology classes, yielding Fourier-transform relations between class counts and spectral data, and they obtain asymptotic formulas for cycle and prime-cycle counts in homology classes, including vanishing phenomena in parity and torsion cases. By connecting Ihara zeta/L-functions to a unitary character framework, the work provides new tools for analyzing cycle distributions on graphs, with potential implications for prime counting in graph-theoretic settings and for understanding spectral behavior of complex unit gain graphs.

Abstract

We address a prime counting problem across the homology classes of a graph, presenting a graph-theoretical Dirichlet-type analogue of the prime number theorem. The main machinery we have developed and employed is a spectral antisymmetry theorem, revealing that the spectra of the twisted graph adjacency matrices have an antisymmetric distribution over the character group of the graph with a special character called the canonical character being an extremum. Additionally, we derive some trace formulas based on the twisted adjacency matrices as part of our analysis.
Paper Structure (16 sections, 26 theorems, 55 equations, 4 figures)

This paper contains 16 sections, 26 theorems, 55 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Our contributions and paper organization
  4. Preliminaries: orthogonal decomposition, characters, twisted adjacency matrices and L-functions
  5. Orthogonal decomposition
  6. The character group
  7. Twisted adjacency matrices and twisted edge adjacency matrices
  8. Basic properties of the L-functions
  9. The Ihara determinant formula and computation of the L-functions
  10. Main results
  11. The canonical character and spectral antisymmetry
  12. Some trace formulas
  13. Counting cycles in homology classes
  14. An example on distributions of spectral radii and trace function
  15. An example on trace formulas and cycle counting
  16. ...and 1 more sections

Key Result

Lemma 2.2

$\omega=\sum_{{\mathbf e}\in {\mathbf E}_O(G)} \omega_{\mathbf e}\cdot d{\mathbf e}$ is a harmonic $1$-form if and only if $d^*\omega = 0$ if and only if $\sum_{{\mathbf e}\in {\mathbf E}_O(G),{\mathbf e}(1)=v}\omega_{\mathbf e} = \sum _{{\mathbf e}\in {\mathbf E}_O(G),{\mathbf e}(0)=v}\omega_{\mat

Figures (4)

  • Figure 1: A genus-$2$ graph: two vertices $v$ and $w$ are connected by three paths $\Delta_0$, $\Delta_1$ and $\Delta_2$ of length $l_0$, $l_1$, and $l_2$ respectively.
  • Figure 2: Distributions of spectral radii $\rho(W_{1,\omega})$ over the character groups of graphs (a) $G_1$ with $l_0=1, l_1=2, l_2=3$, (b) $G_2$ with $l_0=1, l_1=3, l_2=5$ and (c) $G_3$ with $l_0=2, l_1=2, l_2=4$. For each case, the horizontal and vertial axes correspond to $\phi_1(d{\mathbf e}_1)$ and $\phi_2(d{\mathbf e}_2)$ respectively.
  • Figure 3: Distributions of ${\mathcal{K}}(\omega,l)$ over the character groups of graphs $G_1$, $G_2$ and $G_3$. For each case, the horizontal and vertial axes correspond to $\phi_1(d{\mathbf e}_1)$ and $\phi_2(d{\mathbf e}_2)$ respectively.
  • Figure 4: An example for $K_4$.

Theorems & Definitions (67)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3: Hodge orthogonal decomposition
  • proof
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Remark 2.6
  • Proposition 2.7
  • ...and 57 more