Spectral and Extremal Graph Theory with Applications

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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.