Textbook in PDF format Divided into twelve chapters, this volume is an introduction to Graph Theory and Combinatorics and their Applications. It presents its content in a simple way, and contains a wide variety of applications to real world Science and Engineering problems. Definitions and theories are discussed with the help of examples. At the end of each chapter, a revision section is included, which incorporates multiple choice questions and will be highly useful for students undergoing competitive exams. Acknowledgement. Preliminary. Introduction Eulerian and Hamiltonian Graphs. Trees. Spanning Trees and Optimization. Flow Networks. Planarity and Matching Theory. Graph Coloring and Chromatic Polynomials. Counting Techniques. Inclusion and Exclusion. Generating Functions. Recurrence Relations. Rook Polynomials and Combinatorial Designs