Application of quotient graph theory to three-edge star graphs

TL;DR

This paper applies quotient graph theory to symmetric three-edge star quantum graphs, deriving smaller quotient graphs that are unitarily equivalent to the original, for various boundary and coupling conditions.

Contribution

It extends quotient graph theory to three-edge star graphs with specific symmetries and boundary conditions, demonstrating their unitary equivalence to the original Hamiltonian.

Findings

Derived quotient graphs for symmetric star graphs.

Proved unitary equivalence between quotient and original Hamiltonians.

Analyzed different coupling conditions at the central vertex.

Abstract

We apply the quotient graph theory described by Band, Berkolaiko, Joyner and Liu to particular graphs symmetric with respect to $S_3$ and $C_3$ symmetry groups. We find the quotient graphs for the three-edge star quantum graph with Neumann boundary conditions at the loose ends and three types of coupling conditions at the central vertex (standard, $\delta$ and preferred-orientation coupling). These quotient graphs are smaller than the original graph and the direct sum of quotient graph Hamiltonians is unitarily equivalent to the original Hamiltonian.