Spectral and Extremal Graph Theory with Applications

格雷厄姆9801446这个项目涉及谱图论、极值图论和准随机图的几个相互关联的领域。其目的是从图的谱或容易计算的不变量的简短列表中推导出图的基本性质和结构。一种方法是检查各种图性质的关系，并识别图性质的大的等价类。用几何方法求出了随机游动问题的谱界，提高了收敛速度。这项研究的一部分是与对与海量数据集中出现的许多组合问题有关的非常大的图的研究紧密结合在一起的。这项研究属于组合学领域。组合学的目标之一是找到有效的方法来操作和列举离散的对象集合。离散系统的行为对于现代通信来说是极其重要的。例如，大型网络的设计，如那些发生在电话系统中的网络，以及计算机科学中的算法设计，都涉及离散的对象集，这利用了组合研究。

Graham 9801446 This project involves several interrelated areas in spectral graph theory, extremal graph theory, and quasi-random graphs. The 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. One approach is to examine the relationship of various graph properties and to identify large equivalence classes of graph properties. Geometric methods are used to derive spectral bounds and to improve the rate of convergence for random walk problems. Part of the research is closely coupled with the study of very large graphs in connection with many combinatorial problems arising in massive data sets. This research is in the area of combinatorics. One of the goals of combinatorics is to find efficient methods for manipulating and enumerating discrete collections of objects. The behavior of discrete systems is extremely important to modern communications. For example, the design of large networks, such as those occurring in telephone systems, and the design of algorithms in computer science deal with discrete sets of objects, and this makes use of combinatorial research.