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In this work, we demonstrate that edge-oxidation of graphene can enable larger enhancement in thermal conductivity (k) of graphene-nanoplatelet (GnP)/polyetherimide (PEI) composites relative to oxidation of the basal plane of graphene. Edge oxidation (EGO) offers the advantage of leaving the basal plane of graphene intact, preserving its high in-plane thermal conductivity (kin > 2000 Wm-1K-1), while, simultaneously, the oxygen groups introduced on graphene edge enhance interfacial thermal conductance through hydrogen bonding with oxygen groups of polyetherimide (PEI), enhancing overall polymer composite thermal conductivity. Edge oxidation is achieved in this work, by oxidizing graphene in presence of sodium chlorate and hydrogen peroxide, introducing carboxyl groups on the edge of graphene. Basal-plane oxidation of graphene (BGO), on the other hand, achieved through Hummers method, distorts sp2 carbon-carbon network of graphene, dramatically lowering its intrinsic thermal conductivity, causing the BGO/PEI composite k to be even lower than pristine GnP/PEI composite k value. The resulting thermal conductivity of EGO/PEI composite is found to be enhanced by 18%, whereas that of BGO/PEI composite is diminished by 57%, with respect to pristine GnP/PEI composite for 10 weight% GnP content. 2-dimensional Raman mapping of graphene nanoplatelets is used to confirm and distinguish the location of oxygen functional groups on graphene. The superior effect of edge bonding presented in this work can lead to fundamentally novel pathways for achieving high thermal conductivity polymer composites.