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"this book is highly recommended to those with an interest in some of the deepest and most beautiful results in modern number theory as well as the applications to the spectral theory of graphs." Mathematics Abstracts "This book is an excellent introduction to modern topics in number theory, with the last chapter devoted to applications to communications networks." Mathematical Reviews Novel and important applications of number theory to graph theory and vice versa had been made in the past decade. 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, such as Weil conjectures, Riemann-Roch theorem, L-functions, character sum estimates, modular forms, and representation theory.