The theme of this book is some chemical topological indices of graphs. The book is divided into nine chapters. Chapter 1 presents some basic terminologies and notations. Chapter 2 is about two conjectures of Randi index. Chapter 3 devotes to extremal Narumi-Katayama index of graphs with given conditions. Chapter 4 provides some extremal results on the ABC index with degree sequence, matching and pendent vertices among trees and unicyclic graphs. Chapter 5 discusses the comparison between Winner index and Zagreb indices. The weighted vertex PI index and vertex PI index of unicyclic graphs are studied in Chapter 6. In Chapter 7, the smallest Hosoya index of bicyclic graphs with pendent vertices are given. Chapter 8 presents the maximum Sum-Balaban index of tree with given diameter and some open problems. Chapter 9 discusses the irregularity of graphs.