Basic properties of some matrices associated with a graph

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 translate the second, third and fourth chapters of the mentioned book into Turkish in order to understand and explain the "basic properties of some matrices associated with a graph", 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 by using the first chapter of the aforementioned book and the first and the second chapters of Nica (2018). In addition to these, the articles listed in the references were also consulted. To summarize in outline: In the second chapter, the properties of the incidence matrices of directed and undirected graphs are studied; moreover, a method for finding the Moore-Penrose inverse of an incidence matrix is given; the relationship between incidence matrices and path matrices is revealed and the König-Egervary theorem is proven. In the third chapter, the properties of an adjacency matrix are studied; eigenvalues of (adjacency matrices of) some graphs are calculated, some bounds related to the largest eigenvalue of a graph are determined, and a lower bound for the chromatic number of a graph is found by using the smallest and largest eigenvalues of the graph; furthermore, anti-adjacency matrix of a directed graph is defined and the properties of adjacency matrices of regular trees are examined. In the fourth chapter, the properties of a Laplacian matrix are studied; Laplace eigenvalues of some graphs are computed, and bounds are determined for the Laplace spectral radius; in addition, the Matrix-Tree theorem is proven, and the edge-Laplacian matrix of a tree is defined and the Moore-Penrose inverse of it is determined. In the fifth chapter of the thesis, the last chapter 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.