Many reported polymer/graphene composites, even some with excellent graphene dispersion, do not possess impressive conductivity, which may be due to the failure in forming interconnected conductive paths. In this work, a kind of bio-based polyester (BE) is synthesized by polycondensation between plant-derived diols and diacids. It is then grafted onto graphene oxide (GO) via the easterification between hydroxyls of BE and carboxyls of GO. Subsequently, the grafted GO is subjected to reduce by vitamin C. Because of the presence of terminal hydroxyl group in both ends of the BE chains and multiple carboxyl groups on GO, the grafts form specific spatial interconnection structure in BE matrix. The resulting BE/graphene composites possess impressive low threshold percolation of electrical conductivity, in combination of high thermal conductivity.