Reducibility of Cartesian product quantum graph equipped with group action

Abstract

We consider a Cartesian product quantum graph ${\Gamma }_{n_1}\square {\Gamma }_{n_2}$ with standard vertex conditions, and complete the decomposition of Hilbert space $L^2({\Gamma }_{n_1}\square {\Gamma }_{n_2})$ and the Laplacian $\mathcal{H}$ on it by employing the relevant theories of group representation. The concept of ${\Gamma }_{n_1}\square {\Gamma }_{n_2}$ 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 $C_{n_1{n}_2}(n_1,n_2)$. This work also provides a new method for the construction of isospectral graphs.

14 pages,7 figures,1 table