A nano-scale strip of graphene is known as graphene nano-ribbon (GNR). Previous studies have shown that the armchair-type GNR (aGNR) can open the electronic band gap at room temperature, and the band gap increases monotonically with the decrease in the width of aGNR. The critical width at which aGNR shows semi-conductive characteristics at room temperature is about 70 nm, when it is passivated by hydrogen on both sides. However, the electronic band structure varies frequently as a function of the number of carbon atoms along its width direction. In order to decrease the large variation of the band gap of aGNR to control the electronic properties of GNR for highly sensitive sensors and high performance devices, the electronic band structure of various dumbbell-shape structure of aGNR was analyzed by first-principles calculations based on the density functional theory using implemented in SIESTA package. It was shown that the width of aGNR had a large effect on the electronic band structure and the amplitude of the fluctuation of the band gap as a function of the number of carbon atoms decreased drastically. The electronic band structure of various GNRs under the application of uniaxial strain was also analyzed by using the first-principles calculations, in this study. It was confirmed that the effective band gap of aGNR thinner than 70 nm varies drastically under the application of uniaxial strain, and this result clearly indicates the possibility of a highly sensitive strain sensor using dumbbell-shape GNR structures.