We study narrow zigzag graphene nanoribbons (ZGNRs), employing density functional theory (DFT) simulations and the tight-binding (TB) method. The main result of these calculations is the braiding of the conduction and valence bands, generating Dirac cones for noncommensurate wave-vectors k- . Employing a TB Hamiltonian, we show that the braiding is generated by the third-neighbor hopping. We calculate the band structure, the density of states, and the conductance; new conductance channels are opened, and the conductance at the Fermi energy assumes integer multiples of the quantum conductance unit Go=2e2/h. We also investigate the satisfaction of the Stoner criterion by these ZGNRs. We calculate the magnetic properties of the fundamental state, employing the random-phase approximation and employing local spin-density approximation (LSDA) (spin-unrestricted DFT) we confirm that ZGNRs with N=(2,3) do not satisfy the Stoner criterion and as such, the magnetic order could not be developed at their edges. These results are confirmed by both tight-binding and LSDA calculations.