Recent advancements in large-scale pretraining have demonstrated significant potential for learning universal representations that can be used for downstream tasks. However, capturing and transferring structural information across different graph domains remains challenging, primarily due to inherent differences in topological patterns within various contexts. Furthermore, most existing models struggle to capture the complexity of rich graph structures, leading to insufficient exploration of the embedding space. To address these challenges, we introduce GFSE, a Generalized Flexible Structural Encoder designed to capture transferable structural patterns across multiple domains, such as molecular graphs, social networks, and citation networks. GFSE represents a groundbreaking approach as the first cross-domain graph structural encoder to utilize multiple self-supervised learning objectives during pretraining. Leveraging Graph Transformers, GFSE enhances its attention mechanism with graph-induced biases, enabling it to encode complex multi-level and fine-grained topological features effectively. The pretraining process of GFSE generates highly versatile and theoretically powerful positional and structural encodings. These encodings can be seamlessly integrated into various downstream graph feature encoders, including graph neural networks for vectorizing features and large language models for attribute-rich text graphs. Comprehensive experiments on both synthetic and real-world datasets highlight the model's ability to significantly boost performance while substantially reducing the need for task-specific fine-tuning. Notably, GFSE achieved state-of-the-art performance in 81.6% of the evaluation cases, spanning a diverse range of graph models and datasets. This achievement underscores GFSE's potential as a robust and multifunctional tool for encoding graph structural data, making it a valuable addition to the field of graph machine learning.