Reducibility of Cartesian product quantum graph equipped with group action

TL;DR

This paper analyzes the spectral properties of Cartesian product quantum graphs with group actions, decomposing their Hilbert space and Laplacian using group representation theory, and introduces methods for constructing isospectral graphs.

Contribution

It provides a detailed decomposition of the Hilbert space and Laplacian for quantum graphs with group actions, and presents a new approach to constructing isospectral graphs.

Findings

Decomposition of Hilbert space and Laplacian using group representation theory.

Construction of quotient graphs and secular determinant decomposition.

Equivalence of certain quantum graphs to circulant graphs when gcd condition holds.

Abstract

We consider a Cartesian product quantum graph $\Gamma_{n_1}\Box\Gamma_{n_2}$ with standard vertex conditions, and complete the decomposition of Hilbert space $L^2(\Gamma_{n_1}\Box\Gamma_{n_2})$ and the Laplacian $\mathscr{H}$ on it by employing the relevant theories of group representation. The concept of $\Gamma_{n_1}\Box\Gamma_{n_2}$ 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.