  The electronic nonlinear transport through ultra narrow graphene nanoribbons
(sub-$10nm$) is studied. A stable region of negative differential resistance
(NDR) appears in the I-V characteristic curve of {\it odd} zigzag graphene
nanoribbons (ZGNRs) in both positive and negative polarity. This NDR originates
from a transport gap inducing by a selection rule which blocks electron
transition between disconnecting energy bands of ZGNR. Based on this transition
rule, on/off ratio of the current increases exponentially with the ribbon
length up to $10^5$. In addition, charging effects and also spatial
distribution of bond currents was studied by using non-equilibrium Green's
function formalism in the presence of e-e interaction at a mean field level. On
the other hand, we also performed an {\it ab initio} density functional theory
calculation of transmission through a passivated graphene nanoribbon to
demonstrate robustness of the transport gap against hydrogen termination of the
zigzag edges.

Mesoscale and Nanoscale Physics