The Harary Index of a Graph This is the first book to focus on the topological index the Harary index of a graph including its mathematical properties chemical applications and some related and attractive open problems. This book is dedicated to Professor Frank Harary (1921 2005) the grandmaster of graph theory and its applications. It has be written by experts in the field of graph theory and its applications. For a connected graph G as an important distance based topological index the Harary index H(G) is defined as the sum of the reciprocals of the distance between any two unordered vertices of the graph G. In this book the authors report on the newest results on the Harary index of a graph. These results mainly concern external graphs with respect to the Harary index; the relations to other topological indices; its properties and applications to pure graph theory and chemical graph theory; and two significant variants i.e. additively and multiplicatively weighted Harary indices. In the last chapter we present a number of open problems related to the Harary index. As such the book will not only be of interest to graph researchers but to mathematical chemists as well. Introduction Extremal Graphs with Respect to Harary Index Relation Between the Harary Index and Related Topological Indices Some Properties and Applications of Harary Index The Variants of Harary Index Open Problems