This book includes a number of research topics in graph theory and its applications. We discuss various research ideas devoted to alpha-discrepancy, strongly perfect graphs, the reconstruction conjectures, graph invariants, hereditary classes of graphs, embedding graphs on topological surfaces, as well as applications of graph theory, such as transport networks and hazard assessments based on unified networks. The book has a free-form structure that allows the reader freedom, that is, the chapters are independent and can be read in any order. This book is ideal for developers of grant proposals, as well as for researchers interested in exploring new areas of graph theory and its applications. Advanced students in graph theory may use the topics presented in this book to develop their final-year projects, master's theses or doctoral dissertations. It is the author's hope that this publication of original research ideas, problems and conjectures will instigate further research, or even a resurgence of interest, in the aforementioned important areas of graph theory.