Abstract: Cross-domain Sequential Recommendation (CSR) aims to capture the behavioral preferences of users by modeling their historical interaction sequences in multiple domains, thus providing personalized cross-domain recommendations. Recently, researchers have started integrating Graph Convolution Networks (GCNs) into CSR to model complicated associations among users and items. However, due to their complicated structure, most graph-based CSR methods are usually accompanied by high computational complexity or memory overhead, making them difficult to deploy on resource-constrained edge devices. Besides, existing lightweight graph-based CSR methods tend to employ Single-Layer Aggregating Protocol (SLAP) to propagate embeddings on Cross-domain Sequential Graphs (CSG). Such a strategy indeed aids the GCNs in circumventing cross-domain noise interference caused by high-order neighborhood aggregation strategies. However, it also shields GCN from mining high-order sequential relationships within individual domains. To this end, we introduce a lightweight Tri-branches graph External Attention Network (TEA-Net). Specifically, we separate the original CSG into three parts (i.e., two inner-domain sequential graphs and an inter-domain sequential graph) and devise a parallel tri-branches graph convolution network to learn the node representations. This structure can simultaneously consider the first-order inter-domain correlations and the high-order inner-domain connectivity without introducing additional cross-domain noises. Besides, we propose an improved External Attention (EA) component without the nonlinear channel, which captures the sequential dependency among items at a lower cost and shares attention weights across multiple branches. We conduct extensive experiments on two large-scale real-world datasets to verify the performance of TEA-Net. The experimental results demonstrate the superiority of TEA-Net in both the lightweight performance and the prediction accuracy compared to several state-of-the-art methods.