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With growing concern about environmental issues and an emerging green communications paradigm, cognitive radio (CR) networks (CRNs) have to be considered from the energy efficiency perspective. In this paper, we focus on scheduling in CRNs, in which a cognitive base station (CBS) makes frequency allocations to the CRs at the beginning of each frame. A cognitive scheduler must consider the diversity among CRs' queues and channel capacities in terms of number of bits and the channel switching cost from one frequency to another. Taking all these into account, we formulate the scheduling problem as an energy efficiency maximization problem, which is a nonlinear integer programming (NLP) problem and is thereby hard to solve. We seek alternate computationally easier solutions. To this aim, we propose a polynomial-time heuristic algorithm, i.e., the energy-efficient heuristic scheduler (EEHS), which allocates each idle frequency to the CR that attains the highest energy efficiency at this frequency. Next, we reformulate the original problem first as a throughput maximization problem subject to energy consumption restrictions and then as an energy consumption minimization problem subject to minimum throughput guarantees. These two schedulers also have the power to provide fairness in resource allocation. We analyze the energy efficiency and successful transmission probability of the proposed schedulers under both contiguous and fragmented spectrum scenarios. Performance studies show that, compared with a pure opportunistic scheduler with a throughput maximization objective, proposed schedulers can attain almost the same throughput performance with better energy efficiency.