Energy harvesting (EH) technology is an effective way to resolve the energy supply problem for green Internet of Things application. However, it is extremely vulnerable to the unpredictable environmental changes, and as a result, sensor nodes are charged in an uncontrollable way. In this paper, we focus on EH cooperative wireless sensor networks (EHC-WSNs), a new type of WSNs that integrates EH and wireless energy transfer technologies to provide the continuous and controllable energy supply. We propose the graphene-grid deployment strategy to guarantee energy coverage and network connectivity, whereby a Graphene-based Energy Cooperation Management (GECM) mechanism is designed under the energy-neutral operation. Furthermore, we divide GECM into two different phases, i.e., graphene-based energy cooperative charging strategy and graphene-based opportunistic cooperative routing algorithm, which are optimized according to the graphene-grid structure. Extensive simulations show that the proposed GECM can maximize the harvested energy utilization and prolong the network lifetime.