Zero-energy states in triangular and trapezoidal graphene structures

We derive analytical solutions for the zero-energy states of degenerate shell obtained as a singular eigen-value 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 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.