Deep neural networks (DNNs) have substantial computational and memory requirements, and the compilation of its computational graphs has a great impact on the performance of resource-constrained (e.g., computation, I/O, and memory-bound) edge computing systems. While efficient execution of their computational graph requires an effective scheduling algorithm, generating the optimal scheduling solution is a challenging NP-hard problem. Furthermore, the complexity of scheduling DNN computational graphs will further increase on pipelined multi-core systems considering memory communication cost, as well as the increasing size of DNNs. Using the synthetic graph for the training dataset, this work presents a reinforcement learning (RL) based scheduling framework RESPECT, which learns the behaviors of optimal optimization algorithms and generates near-optimal scheduling results with short solving runtime overhead. Our framework has demonstrated up to $\sim2.5\times$ real-world on-chip inference runtime speedups over the commercial compiler with ten popular ImageNet models deployed on the physical Coral Edge TPUs system. Moreover, compared to the exact optimization methods, the proposed RL scheduling improves the scheduling optimization runtime by up to 683$\times$ speedups compared to the commercial compiler and matches the exact optimal solutions with up to 930$\times$ speedups. Finally, we perform a comprehensive generalizability test, which demonstrates RESPECT successfully imitates optimal solving behaviors from small synthetic graphs to large real-world DNNs computational graphs.