To increase mobile batteries' lifetime and improve quality of experience for computation-intensive and latency-sensitive applications, mobile edge computing has received significant interest. Designing energy-efficient mobile edge computing systems requires joint optimization of communication and computation resources. In this paper, we consider energy-efficient resource allocation for a multi-user mobile edge computing system. First, we establish on two computation-efficient models with negligible and non-negligible base station (BS) executing durations, respectively. Then, under each model, we formulate the overall weighted sum energy consumption minimization problem by optimally allocating communication and computation resources. The optimization problem for negligible BS executing duration is convex, and we obtain the optimal solution in closed-form to this problem. The optimization problem for non-negligible BS executing duration is NP-hard in general, and we obtain a sub-optimal solution with low-complexity to this problem, by connecting it to a three-stage flow-shop scheduling problem and wisely utilizing Johnson's algorithm. Finally, numerical results show that the proposed solutions outperform some baseline schemes.