Thadathil, Gifan2018-10-042018-10-042018-05http://udspace.udel.edu/handle/19716/23879Graphs are abstract mathematical objects used to model networks, and spectral graph theory is the sub eld studying matrices, eigenvalues, and eigenvectors associated with graphs. We consider the spectral graph sparsi cation: the problem of construct- ing sparse approximations of dense graphs with respect to spectral characteristics. We provide an exposition of this problem from the ground up. We explore two motivations for this problem: as a means to contains data explosion and as a generalization of expander graphs. From here, we formalize the problem and provide three probabilistic algorithms to construct spectral sparsi ers. This is done with incremental improve- ment to demonstrate the intuitions and proof techniques underlying sparsi cation al- gorithms. The rst algorithm does simple random sampling with xed probability, and the other two algorithms make use of e ective resistances, of which the latter employs a law of large numbers technique.computer science, spectal sparsification, first principlesThesis