  The significant electron-electron interactions that characterize the
{\pi}-electrons of graphene nanoribbons (GNRs) necessitate going beyond
one-electron tight-binding description. Existing theories of electron-electron
interactions in GNRs take into account one electron-one hole interactions
accurately but miss higher order effects. We report highly accurate density
matrix renormalization group (DMRG) calculations of the ground state electronic
structure, the relative energies of the lowest one-photon versus two-photon
excitations and the charge gaps in three narrow graphene nanoribbons (GNRs)
within the correlated Pariser-Parr-Pople model for {\pi}-conjugated systems. We
have employed the symmetrized DMRG method to investigate the zigzag nanoribbon
3-ZGNR and two armchair nanoribbons 6-AGNR and 5-AGNR, respectively. We predict
bulk magnetization of the ground state of 3-ZGNR, and a large spin gap in
6-AGNR in their respective thermodynamic limits. Nonzero charge gaps and
semiconducting behavior, with moderate to large exciting binding energies are
found for all three nanoribbons, in contradiction to the prediction of
tight-binding theory. The lowest two-photon gap in 3-ZGNR vanishes in the
thermodynamic limit, while this gap is smaller than the one-photon gap in
5-AGNR. However, in 6-AGNR the one-photon gap is smaller than the two-photon
gap and it is predicted to be fluorescent.
