Recently, graph-aware Large Language Models (LLMs) have shown promising capabilities in jointly modeling graph-structured data and textual information. Existing approaches typically employ a graph encoder and a frozen LLM to obtain node representations from graph and textual views, followed by node-level alignment to bridge the two modalities. However, such alignment mechanisms primarily focus on node information while overlooking edge-level structures, leading to suboptimal information propagation across views. In this work, we conduct a comprehensive theoretical analysis to uncover why node-level alignment is insufficient for aligning textual and graph representations. Specifically, we prove theoretically for the first time that neglecting edge information leads to suboptimal solutions and negatively curved edges induce bottlenecked information flow, giving rise to the over-squashing phenomenon between graph and textual views. To address the two challenges, we innovatively proposed a CureLLM framework of Curvature-enhanced Graph Representations for Large Language Model whose goal is to inject the signals of edge information into the existing LLMs. Specifically, CureLLM first introduces the training-free textual prompt mechanism to make the LLM model generate the output directly based on the edge-aware prompt without learnable parameter costs. Furthermore, a novel curvature-aware graph representation learning is designed to capture the edge structure information to enhance the downstream tasks, where the message passing between text and graph representations only depends on edges with positive curvature. Finally, we conduct evaluations with 20 different compared methods on 11 real world datasets from various domains and the experiment results demonstrate the superiority of our proposed CureLLM framework.