Graphene, a single atomic layer of $sp^2$-hybridized carbon, produced by chemical vapor deposition (CVD) has immense potential as a transparent conducting material in electronic applications owing to its extraordinary properties including optical transparency, mechanical flexibility and stretchability, especially in next-generation displays. Particularly, graphene-based novel electrodes have the potential to satisfy the important factors for various electronic devices such as thin-film transistors (TFTs) and organic light-emitting diodes (OLEDs). In this thesis, firstly, we investigate the tunable charge transport properties as a function of graphene doping in metal oxide-based TFTs in terms of the electrical characteristics and low-frequency noise (LFN) behaviors for achieving graphene-based high performance devices. The level of contact noise created by the barrier height fluctuation and relative contribution of channel and contact noise in the TFTs was investigated by LFN measurements to demonstrate the tunable charge transport. Secondly, we demonstrate stretchable TFTs consisting of two-dimensional (2D) graphene and molybdenum disulfide on the polymer substrate taking advantage of their excellent mechanical characteristics, and investigate the electrical characteristics of TFTs under mechanical deformation. The mechanisms of performance degradation in TFTs are investigated in the aspect of change in contact resistance closely associated with the relative deformation of 2D materials under mechanical stretching condition. Finally, we propose the novel graphene electrode architecture by applying a selective defect healing technique to CVD-grown graphene, which has achieved essential elements including sheet resistance, transmittance, charge transport, stretchability and flexibility for next generation transparent electrode. Moreover, we demonstrate a graphene-based high-performance OLED device that integrates the proposed electrode architecture on flexible substrates. Therefore, our findings provide new insights into the immense potential of graphene as an advanced transparent conductor toward high-performance stretchable and flexible displays.