During the past decade, graphene --- a monolayer of carbon atoms --- has attracted enormous interest for its use in nanoelectronic device applications. The absence of bandgap, however, has stalled its use both in logic (inability to turn off) and radio frequency (poor power gain) applications. Graphene nanoelectronic devices based on alternative and complementary approaches, which yet exploit its fundamental properties rather than trying to change them, are needed for realistic applications. The work in this thesis proposes two such alternative approaches for graphene's application. The first approach examines the use of graphene as a membrane of radio frequency (RF) nanoelectromechanical systems (NEMS) capacitive switches. Owing to its extreme thinness and exceptional mechanical properties, the use of graphene in RF NEMS switches could enable lower actuation voltages and faster switching. To evaluate its electromagnetic performance, a framework for the full-wave simulation of graphene-based RF NEMS switch is developed for the first time. A rigorous modeling approach for graphene NEMS switch taking into account both its frequency-dependent conductivity, and the variation of conductivity in the up- and down-state is presented. Our results show that RF NEMS switches based on graphene with lower sheet resistivity values can deliver superior isolation and reduced losses at micro and millimeter wave frequencies, and their isolation can also be tuned with bias voltage. An attempt is also made to characterize the fabricated switches. The second approach deals with the negative differential resistance (NDR) phenomenon in planar graphene solid-state devices. The key advantage of planar graphene-based NDR devices is their ability to exhibit NDR at higher current levels, thanks to its high mobility and saturation velocity. The observation of NDR is reported in the output characteristics of graphene field effect transistors for various channel lengths and dielectric thicknesses at room temperature. The transistors are fabricated using chemical vapor deposition graphene with a top gate oxide down to 2.5 nm of equivalent oxide thickness. To understand the NDR phenomenon in graphene transistors, we perform extensive theoretical studies based on drift-diffusion model. This understanding allows us to design a novel graphene circuit which shows enhanced NDR characteristics and is more relevant for applications. Finally, the potential of this graphene NDR circuit is evaluated for RF reflection amplifiers application.