Novel and important applications of number theory to graph theory and vice versa had been made since the mid-1990s. The two main tools used are based on the estimates of character sums and the estimates of the eigenvalues of Hecke operators, both are rooted in the celebrated Weil conjectures settled by Deligne in 1973. The purpose of this book is to give from scratch, a coherent and comprehensive introduction to the topics in number theory related to the central tools, with the ultimate goal of presenting their applications. This book includes many important subjects in number theory. These include Weil conjectures, Riemann-Roch theorem, L-functions, character sum estimates, modular forms, and representation theory.