Pristine graphene is a semimetal and thus does not have a band gap. By making
a nanometer scale periodic array of holes in the graphene sheet a band gap may
form; the size of the gap is controllable by adjusting the parameters of the
lattice. The hole diameter, hole geometry, lattice geometry and the separation
of the holes are parameters that all play an important role in determining the
size of the band gap, which, for technological applications, should be at least
of the order of tenths of an eV. We investigate four different hole
configurations: the rectangular, the triangular, the rotated triangular and the
honeycomb lattice. It is found that the lattice geometry plays a crucial role
for size of the band gap: the triangular arrangement displays always a sizable
gap, while for the other types only particular hole separations lead to a large
gap. This observation is explained using Clar sextet theory, and we find that a
sufficient condition for a large gap is that the number of sextets exceeds one
third of the total number of hexagons in the unit cell. Furthermore, we
investigate non-isosceles triangular structures to probe the sensitivity of the
gap in triangular lattices to small changes in geometry
