The problem of user scheduling and power allocation in full-duplex (FD) cellular networks is considered in this paper, where a FD base station communicates simultaneously with one half-duplex (HD) user on each downlink and uplink channel. First, we propose low-complexity scheduling algorithms to maximize the sum rate of the considered FD system. Second, we derive the optimal power allocation, which is exploited to introduce efficient metrics for FD/HD mode switching in the scheduling to further boost the system rate performance. We analyze the average sum-rate performance of the proposed FD communication algorithms and provide closed-form expressions. Our representative performance evaluation results for the algorithms with and without optimal power control offer useful insights on the interplay among rate, transmit powers, self-interference (SI) cancellation capability, and available number of users in the system.