Chemical graph theory is a specialized interdisciplinary field that interprets and analyzes the molecular structure of chemical compounds using graphs. By combining elements of mathematics and chemistry, it deepens understanding and finds solutions to complex chemical problems by applying the main results and knowledge encompassed in graph theory, a branch of discrete mathematics. At the beginning of this thesis, by precisely defining the basic mathematical concepts of graph theory and proving their fundamental properties, we create a foundation for studying the chemical world from a mathematical perspective. Transferring the fundamental concepts of graph theory into the chemical environment is very intuitive, creating a powerful and widely applicable tool that we then use to analyze the structures and structure-related properties of chemical compounds. As the simplest example of the concrete use of mathematical concepts and principles of graph theory in chemical graph theory, we provide the proof of the general molecular formulas of alkanes and alkenes. Finally, we explain the concept and use of the topological index as a numerical value associated with various structures of chemical compounds to quantitatively describe their shape and enable further analysis of their behavior. Special attention is given to the study of theWiener and Randi´c indices, as two of the most important and frequently used topological indices. After a detailed study of all the previously mentioned topics, we are convinced that graph theory is indeed an indispensable part of chemistry that has strongly influenced its development. On the other hand, chemical graph theory has certainly become one of the most important practical applications of the mathematical skills and knowledge contained in graph theory.