  The evolution of electronic structure of graphene nanoribbons (GNRs) as a
function of the number of layers stacked together is investigated using
\textit{ab initio} density functional theory (DFT) including interlayer van der
Waals interactions. Multilayer armchair GNRs (AGNRs), similar to single-layer
AGNRs, exhibit three classes of band gaps depending on their width. In zigzag
GNRs (ZGNRs), the geometry relaxation resulting from interlayer interactions
plays a crucial role in determining the magnetic polarization and the band
structure. The antiferromagnetic (AF) interlayer coupling is more stable
compared to the ferromagnetic (FM) interlayer coupling. ZGNRs with the AF
in-layer and AF interlayer coupling have a finite band gap while ZGNRs with the
FM in-layer and AF interlayer coupling do not have a band gap. The ground state
of the bi-layer ZGNR is non-magnetic with a small but finite band gap. The
magnetic ordering is less stable in multilayer ZGNRs compared to single-layer
ZGNRs. The quasipartcle GW corrections are smaller for bilayer GNRs compared to
single-layer GNRs because of the reduced Coulomb effects in bilayer GNRs
compared to single-layer GNRs.
