Graph theory has experienced a tremendous growth during the 20th century. One of the main reasons for this phenomenon is the applicability of graph theory in other disciplines such as physics, chemistry, psychology, sociology, and theoretical computer science. This book aims to provide a solid background in the basic topics of graph theory. It covers Dirac''s theorem on k-connected graphs, Harary-Nashwilliam''s theorem on the hamiltonicity of line graphs, Toida-McKee''s characterization of Eulerian graphs, the Tutte matrix of a graph, Fournier''s proof of Kuratowski''s theorem on planar graphs, the proof of the nonhamiltonicity of the Tutte graph on 46 vertices and a concrete application of triangulated graphs. The book does not presuppose deep knowledge of any branch of mathematics, but requires only the basics of mathematics. It can be used in an advanced undergraduate course or a beginning graduate course in graph theory.