  In neutral graphene dots the Fermi level coincides with the Dirac points. We
have investigated in the presence of a magnetic field several unusual
properties of single electron states near the Fermi level of such a
rectangular-shaped graphene dot with two zigzag and two armchair edges. We find
that a quasi-degenerate level forms near zero energy and the number of states
in this level can be tuned by the magnetic field. The wavefunctions of states
in this level are all peaked on the zigzag edges with or without some weight
inside the dot. Some of these states are magnetic field-independent surface
states while the others are field-dependent. We have found a scaling result
from which the number of magnetic field-dependent states of large dots can be
inferred from those of smaller dots.
