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.