Algebraic connectivity and resistance distance in graph theory

The main source used in the preparation of this thesis is the book by "Graphs and Matrices", by Bapat (2014). Essentially, what we do is to translate the eighth, ninth and tenth chapters of the mentioned book into Turkish in order to understand and explain the concepts of "algebraic connectivity and resistance distance in graph theory", as the title of the thesis suggests. But of course, no literal translation was made; almost every proof was written in more detail, and some parts of the book that were left to the reader were explained and the subject was presented more understandably. In addition, the basic definitions and theorems of matrix theory and graph theory, which are prerequisites for understanding the mentioned chapters, are given in the first chapter of the thesis, using the first chapter of the aforementioned book and the first and second chapters of Nica (2018). In addition to these, the articles listed in the references were also consulted. To summarize: In the second chapter, the subject of algebraic connectivity is discussed in detail, a classification of trees is presented related to this concept, and some lower and upper bounds for algebraic connectivity are given. In the third chapter, definitions and theorems about a distance matrix of a graph are given, distance and Laplace matrices of a tree are discussed, and characteristic values of the distance matrix of a tree are studied. In the first section of the fourth chapter, the concept of resistance distance is introduced and it is shown that it is indeed a distance function; in the second section, the subject of network flows is discussed, and in the last section, the basic properties of the resistance matrix are examined. In the fifth chapter of the thesis, before the conclusions and recommendations chapter, a brief literature review on the subject of the thesis is presented by using the notes at the end of the relevant chapters of the mentioned book.