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In this paper, we investigate device-to-device (D2D) overlaying communications with spectrum-power trading, where D2D users (DUs) consume transmit power to relay cell-edge cellular users (CUs) for uplink transmission in exchange for bandwidth from CUs for D2D communications. The proposed spectrum-power trading aims at exploiting individual disparities from both the spectrum and the power perspectives. Recently, energy efficiency (EE) defined by the ratio of the date rate to the power consumption has become increasingly important for devices due to their limited capacity batteries. As such, our goal is to maximize the weighted sum EE (WSEE) of DUs via a joint D2D relay selection, bandwidth allocation, and power allocation while guaranteeing the quality of service of each CU. Specifically, we study WSEE maximization problems for two different cases, i.e., public-interest DUs and self-interest DUs …