Sombor index and Sombor energy

Graph theory, which started in 1736 with the solution of the mathematical problem called "The seven bridges of Königsberg" by Leonhard Euler, is used in many relational structures from financial and communication systems to social and biological processes in the real world. Graph theory has become useful and functional in maps, navigation, electrical circuits, artificial intelligence, computer networks, and in many structures such as the structure of atomic bonds. For this reason, many articles in the literature try to develop mathematical methods using graph theory. In this thesis, it is aimed to gain a new perspective to the studies by obtaining new mathematical approaches with the help of graph types and matrices for graph convolution networks within the scope of deep learning. In this sense, it would be a correct approach to consider the Sombor index, which is one of the chemical uses of graph theory. Sombor is a city located in Serbia and the Sombor index is a molecular structure found by Gutman in 2021. The Sombor index, which is mostly functional in the molecular structure of atoms, is fairly new and a popular index. The Sombor index is found with the help of the degrees determined by the number of edges emerging from the points of the graph. In this thesis, it is aimed to give a different perspective to the Sombor Index and to obtain important limits. In addition, Sombor Energy, which is the sum of the eigenvalues of the Sombor matrix, will be investigated. It is among the goals to contribute to the literature by studying the Sombor energy, which is defined as the ability of the molecule to do work. In the articles written about graph theory, which ways are used, which mathematical operations are performed while an equation is found, and which lemmas are used in which equations are examined in detail. It is aimed to produce new inequalities by making use of the observations obtained as a result of the examination. It is thought that these inequalities can be used as a useful resource for students who will study in the field of Graph theory and Chemical Graph theory.