Driver drowsiness electroencephalography (EEG) signal monitoring can timely alert drivers of their drowsiness status, thereby reducing the probability of traffic accidents. Graph convolutional networks (GCNs) have shown significant advancements in processing the non-stationary, time-varying, and non-Euclidean nature of EEG signals. However, the existing single-channel EEG adjacency graph construction process lacks interpretability, which hinders the ability of GCNs to effectively extract adjacency graph features, thus affecting the performance of drowsiness monitoring. To address this issue, we propose an edge-end lightweight dual graph convolutional network (LDGCN). Specifically, we are the first to incorporate neurophysiological knowledge to design a Baseline Drowsiness Status Adjacency Graph (BDSAG), which characterizes driver drowsiness status. Additionally, to express more features within limited EEG data, we introduce the Augmented Graph-level Module (AGM). This module captures global and local information at the graph level, ensuring that BDSAG features remain intact while enhancing effective feature expression capability. Furthermore, to deploy our method on the fourth-generation Raspberry Pi, we utilize Adaptive Pruning Optimization (APO) on both channels and neurons, reducing inference latency by almost half. Experiments on benchmark datasets demonstrate that LDGCN offers the best trade-off between monitoring performance and hardware resource utilization compared to existing state-of-the-art algorithms. All our source code can be found at https://github.com/BryantDom/Driver-Drowsiness-Monitoring.