To achieve polymer-graphene nanocomposites with high thermal conductivity (k), it is critically important to achieve efficient thermal coupling between graphene and the surrounding polymer matrix through effective functionalization schemes. In this work, we demonstrate that edge-functionalization of graphene nanoplatelets (GnPs) can enable a larger enhancement of effective thermal conductivity in polymer-graphene nanocomposites relative to basal plane functionalization. Effective thermal conductivity for the edge case is predicted, through molecular dynamics simulations, to be up to 48% higher relative to basal plane bonding for 35 wt% graphene loading with 10 layer thick nanoplatelets. The beneficial effect of edge bonding is related to the anisotropy of thermal transport in graphene, involving very high in-plane thermal conductivity (∼2000 W m-1 K-1) compared to the low out-of-plane thermal conductivity (∼10 W m-1 K-1). Likewise, in multilayer graphene nanoplatelets (GnPs), the thermal conductivity across the layers is even lower due to the weak van der Waals bonding between each pair of layers. Edge functionalization couples the polymer chains to the high in-plane thermal conduction pathway of graphene, thus leading to overall high thermal conductivity of the composite. Basal-plane functionalization, however, lowers the thermal resistance between the polymer and the surface graphene sheets of the nanoplatelet only, causing the heat conduction through inner layers to be less efficient, thus resulting in the basal plane scheme to be outperformed by the edge scheme. The present study enables fundamentally novel pathways for achieving high thermal conductivity polymer nanocomposites.