评估大型语言模型在图论已解决和未解决问题上的表现：对计算机教育的启示

Large Language Models (LLMs) are increasingly utilized by students to explore advanced computer science topics, including graph theory. As these tools become integrated into undergraduate and graduate coursework, understanding their reliability in supporting mathematically rigorous thinking is crucial. A study investigates the performance of an LLM on two related graph theoretic problems: a solved problem concerning the gracefulness of line graphs and an open problem for which no solution is currently known. An eight-stage evaluation protocol, designed to mirror authentic mathematical inquiry, was employed. This protocol encompassed interpretation, exploration, strategy formation, and proof construction. The LLM demonstrated strong performance on the solved problem, successfully generating correct definitions and proofs. However, it struggled to make meaningful progress on the unsolved problem, failing to provide novel insights or valid approaches. This research highlights the LLM's proficiency in handling established mathematical problems while exposing its limitations when confronted with open-ended challenges requiring original thought. The findings offer significant implications for the integration and application of LLMs within computer science education, emphasizing the need for educators to understand both the capabilities and constraints of these AI tools in fostering advanced mathematical reasoning.