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.

1 Introduction. . . . . . . . . . . . . . . . . . . . 1

1.1 Short Introduction to Graph Theory . . . . . . . . .. 1

1.2 Distance in Graphs. . . . . . . . . . . . . . . . . 2

1.3 Harary Index of a Graph. . . . . . . . . . . . . . . . 2

1.4 Harary Matrix of a Graph . . . . . . . . . . . . . . . 4

1.5 Modified Harary Index . . . . . . . . . . . . . . . . . . . . 8

2 Extremal Graphs with Respect to Harary Index . . . . . 13

2.1 General Graphs . . . . . . . . . . . . . . . . . . . . .. . . . . 13

2.2 Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 Generalized Trees . . . . . . . . . . . . . . . . . . . . . 23

3 Relation Between the Harary Index and Related Topological Indices. . 27

3.1 Relation Between the Harary Index and Reciprocal Wiener Index . 27

3.2 Relation Between the Harary Index and Zagreb Indices . . . . . . . 31

4 Some Properties and Applications of Harary Index . . . . . . 35

4.1 Some Properties of Harary Index . . . . . . . . . . . 35

4.2 Application of Harary Index in Pure Graph Theory . . . . . . 39

4.3 Application of Harary Index in Mathematical Chemistry . . . . 40

4.4 Application of Harary Index to Structure–Property Modeling. . . . 48

5 The Variants of Harary Index . . . . . . . . . . . . . . . 55

5.1 Extremal Graphs with Respect to HA and HM . . . . . . . .. 56

5.2 Some Properties of Additively Weighted Harary Index . . . . 63

6 Open Problems . . . . . . . . . . . . . . . . . 69

6.1 Determining the Minimal Harary Index in a Given Set . . 69

6.2 Other Attractive Open Problems . . . . . . 70

**Book Reviews – The Harary Index of a Graph**

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