We studied the problem of optimizing the network topology based on indices such as clustering coefficient, algebraic connectivity and average shortest path length. Not only some properties of the networks having optimal or locally optimal topologies were revealed by theoretical analysis, but also some algorithms that can generate networks with nearly optimal topologies were developed. We also studied some dynamics related problems such as the decentralized estimation of the algebraic connectivity, the convergence analysis of discrete-time recurrent neural networks, the analysis of the number of DC operating points in a certain nonlinear circuits, and the global convergence of iterative solution methods for nonnegative matrix factorization, and obtained many important results through both theoretical analysis and numerical experiments.