graph theory students' perceptions of graph isomorphism

Objective: graph isomorphism is a central topic in graph theory, yet students often struggle to grasp its underlying principles. this study aimed to explore students' perceptions of graph isomorphism and identify the conceptual metaphors they employ when reasoning about this concept. methods: using a two-stage qualitative research design, the study combined surveys and task-based interviews to collect data. the variation theory framework (marton & booth, 2013) was applied to analyze students' understanding and identify critical and superficial features in their reasoning. results: the findings revealed that students frequently focus on superficial features, such as the equality of the number of vertices and edges, while overlooking critical features like bijective attributes. additionally, the conceptual metaphors used by students were categorized into three main themes: sameness, sameness/mapping, and formal definition. these metaphors reflect varying levels of conceptual understanding, with some students relying on intuitive notions of similarity and others employing more formal, mathematical definitions. conclusions: this study highlights the challenges students face in understanding graph isomorphism and underscores the importance of addressing both superficial and critical features in teaching. by identifying the conceptual metaphors students use, educators can develop targeted instructional strategies to bridge gaps in understanding and promote deeper conceptual learning.