In university admissions, interaction networks naturally emerge between prospective students and available majors. Discovering network patterns in this context is often difficult as many existing models require exhaustive data of both the students and majors. In this work, we uncover the patterns by harnessing Graph Convolutional Networks (GCN) to construct a compact vector space, leveraging only the graph's adjacency data to provide applicants and majors embedding representation. These embeddings are then applied to facilitate major recommendation and clustering tasks. For major recommendation, the GCN embeddings demonstrate superior performance over both classical and neural embeddings, achieving improvements of up to 61.06% and 12.17% across smaller (dimension 40) and larger embedding sizes (dimension 80), respectively. This underscores the GCN efficacy in leveraging lower-dimensional representations more effectively than traditional methods. Furthermore, hierarchical clustering based on embedding distances uncovers implicit patterns in applicant preferences based on the fields of study and geographic locations of majors, even in the absence of explicit data on these attributes during training. This highlights the method's capability to derive meaningful insights from interaction networks without exhaustive entity-level data.