Graphene has emerged as a promising material for active plasmonic devices in the mid-infrared (MIR) region owing to its fast tunability, strong mode confinement, and long-lived collective excitation. In order to realize on-chip graphene plasmonics, several types of graphene plasmonic waveguides (GPWGs) have been investigated and most of them are with graphene ribbons suffering from the pattern-caused edge effect. Here we propose a novel nanoplasmonic waveguide with a pattern-free graphene monolayer on the top of a nano-trench. It shows that our GPWG with nanoscale light confinement, relatively low loss and slowed group velocity enables a significant modulation on the phase shift as well as the propagation loss over a broad band by simply applying a single low bias voltage, which is very attractive for realizing ultra-small optical modulators and optical switches for the future ultra-dense photonic integrated circuits. The strong light-matter interaction as well as tunable slow light is also of great interest for many applications such as optical nonlinearities.