Dumitriu, IoanaHoffman, ChristopherBrito, Gerandy2017-10-262017-10-262017-10-262017-08Brito_washington_0250E_17725.pdfhttp://hdl.handle.net/1773/40636Thesis (Ph.D.)--University of Washington, 2017-08This thesis concerns to spectral gap of random regular graphs and consists of two main con- tributions. First, we prove that almost all bipartite biregular graphs are almost Ramanujan by providing a tight upper bound for the non trivial eigenvalues of its adjacency operator, proving Alon's Conjecture for this family of graphs. Secondly, we use a spectral algorithm to recover hidden communities in a random network model we call regular stochastic block model. We rely on a technique introduced recently by Massoullie, which we develop here for random regular graphs.application/pdfen-USCC BY-NCcommunity detectionregular graphsspectral analysisspectral gapMathematicsMathematicsThesis