In this paper, we investigate energy-efficient joint subcarrier pairing, subcarrier allocation, and power allocation algorithms for improving the network energy efficiency (EE) in multiuser amplify-and-forward (AF) relay networks while ensuring the desired quality-of-service (QoS) requirement for the users through the concept of “network price.” Further, we introduce a network price paid for the consumed power as a penalty for the achievable sum rate and formulate a resource allocation problem subject to limited transmit power budget and QoS constraints. The formulated problem is a nonconvex binary mixed-integer nonlinear programming (MINLP) problem and it is hard to solve the problem. We then apply a concave lower bound on the pricing-based network utility to transform the problem into a convex one. The dual decomposition method is adopted to propose a $\pounds $-price resource allocation algorithm to find the near-optimal solution. Next, we discuss the optimal utility-price from an EE perspective. Moreover, we rigorously analyze the behavior of the network pricing-based resource allocation in two-user case under different noise operating regimes, and discuss the corresponding strategies for achieving energy-efficient transmission, generating water-filling and channel-reversal approaches. To strike a balance between the computational complexity and the optimality, we propose a low-complexity suboptimal algorithm. Furthermore, we extend the proposed algorithm to maximize the EE of multiuser multirelay full-duplex (FD) relay networks and the relay networks with an eavesdropper. The performance gain of the proposed algorithms is validated through computer simulations.