Large Language Models (LLMs) exhibit strong reasoning capabilities in complex tasks. However, they still struggle with hallucinations and factual errors in knowledge-intensive scenarios like knowledge graph question answering (KGQA). We attribute this to the semantic gap between structured knowledge graphs (KGs) and unstructured queries, caused by inherent differences in their focuses and structures. Existing methods usually employ resource-intensive, non-scalable workflows reasoning on vanilla KGs, but overlook this gap. To address this challenge, we propose a flexible framework, Enrich-on-Graph (EoG), which leverages LLMs' prior knowledge to enrich KGs, bridge the semantic gap between graphs and queries. EoG enables efficient evidence extraction from KGs for precise and robust reasoning, while ensuring low computational costs, scalability, and adaptability across different methods. Furthermore, we propose three graph quality evaluation metrics to analyze query-graph alignment in KGQA task, supported by theoretical validation of our optimization objectives. Extensive experiments on two KGQA benchmark datasets indicate that EoG can effectively generate high-quality KGs and achieve the state-of-the-art performance. Our code and data are available at https://github.com/zjukg/Enrich-on-Graph.

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