Spectral, Extremal & Probabilistic Methods in Graph Theory with Applications to Information Technology

计划中的研究涉及谱图理论、极值图理论和随机图等几个相关领域。 一个主要目标是从图的谱（或从一个简单的可计算的不变量列表）推导出图的基本性质和结构。 各种组合，几何和概率技术正在开发中，用于检查各种图形不变量和属性的关系和行为。 虽然主要目标是提高我们对离散结构的内在特征和基本原理的理解，但这些原理在处理Internet计算和海量数据集中出现的海量图问题时是非常有效和必要的。 特别感兴趣的是研究具有幂律分布的图，这些图为现实网络提供了良好的近似。 只有部分信息的大型动态图的演化过程正在被分析，这导致了具有挑战性的问题和新的研究方向。

The planned research involves several interrelated areas in spectral graph theory, extremal graph theory and random graphs. A main goal is to deduce the fundamental properties and structures of a graph from its graph spectrum (or from a short list of easily computable invariants). Various combinatorial, geometric and probabilistic techniques are being developed for examining the relations and behaviors of various graph invariants and properties. Although the primary objective is to advance our understanding of the intrinsic characteristics and underlying principles that govern discrete structures, such principles are quite effective and essential in dealing with problems involving massive graphs that arise in Internet computing and massive data sets.A number of combinatorial problems on Internet infrastructures are examined, including the modeling and scaling of massive graphs using probabilistic analysis. Of particular interest is the study of graphs with power law distributions that offer good approximations for realistic networks. The evolution process of large dynamic graphs with only partial information is being analyzed and this has led to challenging problems and new research directions.