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In this paper, an energy-efficient full-duplex (FD) unmanned aerial vehicle (UAV) relaying network is proposed, where UAV acts as a mobile relay and assists information exchange between two transceivers. Specifically, the load-carry-and-delivery scheme is applied to positively take advantage of the time-varying channel gain in delay-tolerant networks; meanwhile, the FD communication policy is used to potentially further increase the energy efficiency (EE). In particular, the self-interference channel gains follow the complex Gaussian distribution instead of being constant. The EE is first rigorously derived and then, the optimum flight speed is determined under the information causality constraint to maximize the EE. Numerical results demonstrate that the proposed scheme outperforms the half-duplex as well as static schemes in terms of the EE. In addition, the impact of the self-interference cancellation factor on the …