Some geometrical applications of graph theory

This thesis consists of four chapters. In the first chapter, the history of Graph Theory is presented. Information is given about the process of solving the Königsberg bridge problem, which was laid down by Swedish mathematician Leonhard Euler. In addition, other problems such as the Four Color Problem that emerged in the light of this problem and their applications are also discussed. The second chapter introduces the basic concepts and theories that are the building blocks of Graph Theory. Information is provided about the types of graphs, operations that can be performed on graphs, and graph matrices. In the third chapter; fundamental definitions and examples related to subgraphs, covers in graphs, closed graphs, compactness and continuity in graphs are presented. Following this, eigenvalues and eigenvectors in different graph structures are examined, and examples related to them are given. Geometric properties of graphs are discussed, and geometric interpretations of graphs are made. In the final chapter, the conclusions, recommendations, and evaluations related to our thesis are presented.