  Using first-principles calculations, we explore the electronic and magnetic
properties of graphene nanomesh (GNM), a regular network of large vacancies,
produced either by lithography or nanoimprint. When removing an equal number of
A and B sites of the graphene bipartite lattice, the nanomesh made mostly of
zigzag (armchair) type edges exhibit antiferromagnetic (spin unpolarized)
states. In contrast, in situation of sublattice symmetry breaking, stable
ferri(o)magnetic states are obtained. For hydrogen-passivated nanomesh, the
formation energy is dramatically decreased, and ground state is found to
strongly depend on the vacancies shape and size. For triangular shaped holes,
the obtained net magnetic moments increase with the number difference of
removed A and B sites in agreement with Lieb's theorem for even A+B. For odd
A+B triangular meshes and all cases of non-triangular nanomeshes including the
one with even A+B, Lieb's theorem does not hold anymore which can be partially
attributed to introduction of armchair edges. In addition, large triangular
shaped GNM could be as robust as non-triangular GNMs, providing possible
solution to overcome one of crucial challenges for the sp-magnetism. Finally,
significant exchange splitting values as large as $\sim 0.5$ eV can be obtained
for highly asymmetric structures evidencing the potential of GNM for room
temperature carbon based spintronics. These results demonstrate that a turn
from 0-dimensional graphene nanoflakes throughout 1-dimensional graphene
nanoribbons with zigzag edges to GNM breaks localization of unpaired electrons
and provides deviation from the rules based on Lieb's theorem. Such
delocalization of the electrons leads the switch of the ground state of system
from antiferromagnetic narrow gap insulator discussed for graphene nanoribons
to ferromagnetic or nonmagnetic metal.
