  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.
