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Reducibility of Cartesian product quantum graph equipped with group action

Shimei Li, Kai Zhang, Jia Zhao

TL;DR

This work studies the reducibility of the Laplacian on the Cartesian product quantum graph $Γ_{n_1}\Box Γ_{n_2}$ under a $G_{n_1}\times G_{n_2}$ symmetry by applying finite-group representation theory to decompose the relevant function space and operator. The authors construct a periodic quantum-graph framework, derive a quotient-graph decomposition, and obtain a factorization of the secular determinant into blocks associated with irreducible representations; notably, when $\gcd(n_1,n_2)=1$ the product is equivalent to the circulant graph $C_{n_1n_2}(n_1,n_2)$, simplifying spectral analysis and enabling isospectral-graph constructions. The main contributions are a complete decomposition of $L^2(Γ_{n_1}\Box Γ_{n_2})$ and $\mathscr{H}$, the quotient-graph framework for the secular determinant, and explicit decomposition formulas that connect to circulant graphs in the coprime case. These results provide a practical pathway to study spectra and design isospectral quantum graphs using symmetry reductions.

Abstract

We consider a Cartesian product quantum graph $Γ_{n_1}\BoxΓ_{n_2}$ with standard vertex conditions, and complete the decomposition of Hilbert space $L^2(Γ_{n_1}\BoxΓ_{n_2})$ and the Laplacian $\mathscr{H}$ on it by employing the relevant theories of group representation. The concept of $Γ_{n_1}\BoxΓ_{n_2}$ equipped with the action of the cyclic group $G_{n_1}\times G_{n_2}$ is defined through the introduction of periodic quantum graph and cyclic groups. We also constructed its quotient graph and accomplish the decomposition of its secular determinant. Furthermore, under the condition that $\gcd(n_1,n_2)=1$, it can be regarded as equivalent to a circulant graph $C_{n_1n_2}(n_1,n_2)$. This work also provides a new method for the construction of isospectral graphs.

Reducibility of Cartesian product quantum graph equipped with group action

TL;DR

This work studies the reducibility of the Laplacian on the Cartesian product quantum graph under a symmetry by applying finite-group representation theory to decompose the relevant function space and operator. The authors construct a periodic quantum-graph framework, derive a quotient-graph decomposition, and obtain a factorization of the secular determinant into blocks associated with irreducible representations; notably, when the product is equivalent to the circulant graph , simplifying spectral analysis and enabling isospectral-graph constructions. The main contributions are a complete decomposition of and , the quotient-graph framework for the secular determinant, and explicit decomposition formulas that connect to circulant graphs in the coprime case. These results provide a practical pathway to study spectra and design isospectral quantum graphs using symmetry reductions.

Abstract

We consider a Cartesian product quantum graph with standard vertex conditions, and complete the decomposition of Hilbert space and the Laplacian on it by employing the relevant theories of group representation. The concept of equipped with the action of the cyclic group is defined through the introduction of periodic quantum graph and cyclic groups. We also constructed its quotient graph and accomplish the decomposition of its secular determinant. Furthermore, under the condition that , it can be regarded as equivalent to a circulant graph . This work also provides a new method for the construction of isospectral graphs.

Paper Structure

This paper contains 10 sections, 4 theorems, 61 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quantum graph
  4. Group representation theory
  5. Construction of Cartesian product graph with group action
  6. The decomposition of function space and Laplacian
  7. Secular determinant and its decomposition
  8. The construction of quotient graph
  9. For the case $gcd(n_1,n_2)=1$.
  10. Decomposition of the secular determinant

Key Result

Theorem 4.1

Let $\Gamma_{n_1}\Box\Gamma_{n_2}$ is a Cartesian product graph equipped with the group $G_{n_1}\otimes G_{n_2}$ action. Let $V(\Gamma_{n_1}\Box\Gamma_{n_2})=\left\{v_i\right\}_{i=1}^{n_1n_2}$, the edges connected to each vertex $v_i$ are denoted as $e_{i,1}, e_{i,2},\dots,e_{i,|E_{v_i}|}$. Then the where $\mathcal{F}_{s,t}$ is square-integrable function spaces on $\Gamma_{n_1}\Box\Gamma_{n_2}$.

Figures (7)

  • Figure 3.1: (a)Metric graph $\Gamma_2$ with edge length 1. (b)Metric graph $\Gamma_3$ with edges length 1 and 2. (c)Cartesian product graph $\Gamma_2\Box\Gamma_3$.
  • Figure 3.2: (a)The circulant graph $C_6\left(1,2\right)$ with dummy vertices. (b)The fundamental domain of $C_6\left(1,2\right)$.
  • Figure 4.1: The cartesian graph of the $\Gamma_3$ and $\Gamma_4$.
  • Figure 4.2: (a) The circulant graph $C_{12}(3,4)$. (b) The fundamental domain of $C_{12}(3,4)$.
  • Figure 5.1: (a)The fundamental domain of $\Gamma_{n_1}\Box\Gamma_{n_2}$.(b)The quotient graph of $\Gamma_{n_1}\Box\Gamma_{n_2}$.
  • ...and 2 more figures

Theorems & Definitions (14)

  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • Definition 3.2
  • Theorem 4.1
  • proof
  • Corollary 4.2
  • Example 4.3
  • Theorem 4.4
  • Remark 4.5
  • ...and 4 more