Some limits for Harmonic and Zagreb index

Graph theory, which emerged with Leonhard Euler's thinking on the mathematical problem called "Seven Bridges of Königsberg," has started to be applied in the solution of non-routine problems in many disciplines, such as chemistry, physics, biology, sociology, and astronomy. Consequently, the scientific studies of related disciplines have incorporated the mathematical methods utilized in graph theory. The birth of graph theory and the use of studies obtained with the help of graph types and matrices have led to the discovery of many new mathematical models. These studies have examined the index concept with related mathematical models in graph theory. This thesis consists of three main parts. First, Harmonic and Zagreb indices, which are topological indices used in graph theory, have been investigated. The indices have compared and discovered new bounds. Second, the definitions of the indexes from different perspectives have been made and added new results to these definitions. Finally, the solar systems graph structure has been exemplified and associated with given indexes.