  We derive analytical solutions for the zero-energy states of degenerate shell
obtained as a singular eigenevalue problem found in tight-binding (TB)
Hamiltonian of triangular graphene quantum dots with zigzag edges. These
analytical solutions are in agreement with previous TB and density functional
theory (DFT) results for small graphene triangles and extend to arbitrary size.
We also generalize these solutions to trapezoidal structure which allow us to
study bowtie graphene devices.
