▎ 摘　　要

The density of states of graphene diverge at six M points in the Brillouin zone, known as Van Hove singularities. For a finite graphene structure, such as nanoflake, similar singularities are found to emerge in the energy spectrum as highly degenerate states. We investigate these degenerate states in various graphene nanoflakes and show that the existence of the singularities is strongly dependent on the geometry, size, and even edge profile of the structures. While highly degenerate states are seen for all the hexagonal structures, no singularities can be found for any triangular nanoflake which has even number of carbon rings along each side. We further reveal that the nanoflakes with the Van Hove singularities exhibit very different optical absorption spectrum from those structures where the singularities are absent. More interestingly, we find that these highly degenerate states can survive when the structural symmetry is broken by a vacancy. Depending on its location and the sizes of structures, the defect may even enhance the degeneracy of the states at the singularities. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4764913]