With the fast development of mobile edge computing (MEC), there is an increasing demand for running complex applications on the edge. These complex applications can be represented as workflows where task dependencies are explicitly specified. To achieve better Quality of Service (QoS), computation offloading is widely used in the MEC environment. However, many existing computation offloading strategies only focus on independent computation tasks but overlook the task dependencies. Meanwhile, most of these strategies are based on search algorithms which are often time-consuming and hence not suitable for many delay-sensitive complex applications in MEC. Therefore, a highly efficient graph-based strategy was proposed in our recent work but it can only deal with simple workflow applications with linear (namely sequential) structure. For solving these problems, a novel graphbased strategy is proposed for workflow applications in MEC. Specifically, this strategy can deal with complex workflow applications with nonlinear (viz. parallel, selective and iterative) structures. Meanwhile, the offloading decision plan with the lowest energy consumption of the end-device under deadline constraint can be found by using the graph-based partition technique. We have comprehensively evaluated our strategy on FogWorkflowSim platform for complex workflow applications. Extensive numerical results demonstrate that the end device's energy consumption can be effectively reduced by 7.81% and 9.51% compared with PSO and GA by the proposed strategy. Meanwhile, the strategy running time is 1% and 0.2% of PSO and GA, respectively.