Laplacian eigenvalues of threshold graphs in graph theory

The main source used in the preparation of this thesis is the graduate textbook "Graphs and Matrices", Bapat (2014). Essentially, what we do is to study the eleventh chapter of the mentioned textbook to understand and explain the "Laplacian eigenvalues of threshold graphs", as can be understood from the title of the thesis. But of course, no part of the aforementioned book has been quoted exactly, a study has been put forward with our own words and our own sentences; almost every proof has been written in more detail, and parts of the book that were left to the reader have been explained completely and the subject has been presented more understandably. In addition to these, the sources listed in the references were also consulted. To summarize in outline: In the first chapter, basic notions are given about the majorization and related fundamental facts are proved. In the first section of the second chapter, threshold graphs are defined, and a characterization of threshold graphs according to the Laplacian eigenvalues is stated and proved; in the second section, the concept of Laplacian integral graph is introduced, and as an example, cographs are defined, which determine a class of Laplacian integral graphs containing the class of threshold graphs, also a characterization of spectral integral variation is stated and proved. In the third chapter, the last chapter before the conclusions and recommendation chapter, a brief literature review on the subject of the thesis is presented mainly by using the notes at the end of the eleventh chapter of the mentioned book.