网络的拓扑结构对网络的性能、系统可靠性及费用都有重大影响。本项目充分利用图论、代数、数论、编码等数学知识，研究网络拓扑图的结构性质和优化设计中的如下科学问题：(1)通过引入图论中的方法，研究网络图的各种结构性质，特别是障碍网络的哈密尔顿性质；(2)结合代数的方法，深入研究网络系统的各类拓扑参数(如图的邻接矩阵特征值、拉普拉斯矩阵特征值、无符号拉普拉斯矩阵特征值、expansion rate等)的极值及其极图；(3)利用数论、编码的理论方法，研究网络优化设计及网络编码的可解性问题。本课题还研究与上述问题相关联的各种实用性算法及算法复杂性分析。

The topological structures of networks highly influence the network efficiency, the reliability and the costs of systems. This project studies the following scientific problems in the topological structure properties of networks and their optimal design by effectively using and integrating the mathematical tools in Graph Theory, Algebra, Number Theory and Coding Theory. (1) Structural properties of networks will be studied by introducing methods from Graph Theory, in particular Hamiltonian properties of faulty networks; (2) topological parameters (such as the eigenvalues of the graph adjacent matrices, the eigenvalues of Laplacian matrices, the eigenvalues of signless Laplacian matrices and the expansion rates of graphs) of network systems will be deeply investigated by integrating algebraic methods; (3) the optimal design of networks and the solvability problems of network coding will be studied using Number Theory methods. Computational methods for the problems listed above and their complexity analysis will also be addressed in this project.