This paper presents a classroom proposal that integrates Graph Theory and Linear Algebra in the context of complete bipartite graphs. The study links adjacency and Laplacian matrices to determine the eigenvalues of these graphs, which are applicable to connectivity concepts. Students engage in exploration activities using GeoGebra software, starting from various specific cases, completing tables and questionnaires to identify patterns in the eigenvalues of the adjacency and Laplacian matrices of complete bipartite graphs. The exploration process leads to generalization, abstracting properties from observed patterns and experimentation. This learning experience bridges the concrete and symbolic, and introduces students to research methods. The proposal is framed within the Realistic Mathematics Education (RME) theory, emphasizing the importance of phenomenological exploration, modeling and symbolism, student construction, interaction, and curriculum integration. The use of GeoGebra software is highlighted as a dynamic tool that enhances learning through experimentation, discovery, reflection, and research. The paper outlines the theoretical framework, key concepts in Linear Algebra and Graph Theory, and the methodology for conducting the activities. It also provides detailed instructions for the first stage of exploration using GeoGebra, followed by the elaboration of conjectures and the generalization of these conjectures. The results section discusses the expected outcomes of the activities, including the identification of patterns and the development of mathematical conjectures. The paper concludes by highlighting the advantages of using GeoGebra in teaching and learning, such as promoting autonomous learning, improving motivation, and fostering communication among students. The integration of technology is seen as essential for motivating students and making classes more innovative and engaging.This paper presents a classroom proposal that integrates Graph Theory and Linear Algebra in the context of complete bipartite graphs. The study links adjacency and Laplacian matrices to determine the eigenvalues of these graphs, which are applicable to connectivity concepts. Students engage in exploration activities using GeoGebra software, starting from various specific cases, completing tables and questionnaires to identify patterns in the eigenvalues of the adjacency and Laplacian matrices of complete bipartite graphs. The exploration process leads to generalization, abstracting properties from observed patterns and experimentation. This learning experience bridges the concrete and symbolic, and introduces students to research methods. The proposal is framed within the Realistic Mathematics Education (RME) theory, emphasizing the importance of phenomenological exploration, modeling and symbolism, student construction, interaction, and curriculum integration. The use of GeoGebra software is highlighted as a dynamic tool that enhances learning through experimentation, discovery, reflection, and research. The paper outlines the theoretical framework, key concepts in Linear Algebra and Graph Theory, and the methodology for conducting the activities. It also provides detailed instructions for the first stage of exploration using GeoGebra, followed by the elaboration of conjectures and the generalization of these conjectures. The results section discusses the expected outcomes of the activities, including the identification of patterns and the development of mathematical conjectures. The paper concludes by highlighting the advantages of using GeoGebra in teaching and learning, such as promoting autonomous learning, improving motivation, and fostering communication among students. The integration of technology is seen as essential for motivating students and making classes more innovative and engaging.