Infinite graphs and recursive structures

The main source used in the preparation of this study is Diestel's (2017) "Graph Theory" book. This thesis mainly consists of studying the eighth chapter of the aforementioned book to understand and explain the subject of "infinite graphs". But of course, no part of the mentioned book has been quoted exactly, a study has been put forward with our own words and our own sentences; and on the one hand, some very difficult or technical proofs were skipped, on the other hand, some parts of the book that were left to the reader were explained and the subject was presented more understandably. In addition to these, the articles listed in the references were also consulted. To summarize in outline: In the first chapter, basic definitions and theorems of graph theory, which are the prerequisites for understanding the main subject, are covered by using the first chapter of the aforementioned book. In the first section of the second chapter, the equivalent of the concept of infinity in graph theory, the basic concepts, techniques, and theorems related to it are studied. In the second section, starting from the fact that there are two fundamentally different aspects of infinity in the connected infinite graphs, roads, trees and ends are introduced and the related results are studied. In the third section, answers are sought to the questions of whether there is a universal graph in the class of all countable graphs and which countable graphs are homogeneous. In the fourth and the last section of the second chapter, the issue of defining a family of graphs as recursive, which makes it possible to use induction to prove theorems about infinite graphs, is discussed. In the third chapter of the thesis, before the conclusions and recommendation chapter, a brief literature review on the subject of the thesis is presented by using the notes section of the eighth chapter of the mentioned book.