In recent years, Large Language Models (LLMs) have achieved remarkable success across various domains, sparking interest in extending their capabilities to graph structured data. However, leveraging LLMs for graph data poses significant challenges due to the inherent mismatch between graph and text modalities. Existing approaches primarily rely on two strategies: Graph-to-Text Translation, which describes graph structures in natural language to enable LLMs to process structural information, and Graph-to-Token Conversion, which transforms graphs into sequences of tokens aligned with text tokens. Although these methods have achieved a certain degree of integration between LLMs and graph data, they still struggle to fully capture the complex structure in real-world graphs and fail to provide a global view of the long-range dependencies—the relationships or interactions between nodes that are far apart in the graph. To bridge this gap and enhance LLM’s understanding of the graph structure, we propose Curvature Enhanced Large Language Model (CELLM), a novel architecture that integrates advanced graph structural information with discrete graph curvature to offer a global and geometric perspective. Graph curvature leverages local curvature measurements to derive global insights, such as assessing overall connectivity, identifying bottlenecks, or detecting hubs. This more expressive structural measure can provide LLMs with enhanced capabilities to capture and comprehend complex graph structures. Additionally, we implement a task-specific tuning procedure to further improve the structure understanding within LLMs. Extensive experiments demonstrate the effectiveness of our proposed CELLM across graph-related tasks, highlighting its potential in improving the expressiveness and understanding of LLMs when applied to graph modalities.