Graphs are fundamental in mathematics since they conveniently encode diverse relations and facilitate combinatorial analysis of many theoretical and practical problems. This book is a revised and expanded version of our earlier introduction to graph theory and combinatorics. The text is ideal for a one-semester course at the advanced undergraduate level or beginning graduate level. Recent developments in the theory of signed adjacency matrices involving the proof of the sensitivity conjecture and the theory of Ramanujan graphs have been added to the second edition, along with other interesting topics such as Pick’s theorem on areas of lattice polygons and Graham-Pollak’s work on addressing of graphs.