The polynomial method and quantitative Friedman theorems for the conditioned configuration model of random regular graphs

Abstract

In this thesis, we study the asymptotic behavior of the spectra of adjacency matrices of random regular multigraphs sampled from the configuration model. The Alon-Boppana theorem gives an asymptotic lower bound of 2sqrt(d-1) on the second-largest eigenvalue of the adjacency matrix of a d-regular graph as the number of vertices N goes to infinity. Friedman's theorem asserts that random regular graphs sampled from a variety of different models almost achieve this bound. For both the unconditioned configuration model and the configuration model with conditioning on the absence of a forbidden subgraph, we apply the polynomial method developed in Chen et al. (2026) to derive quantitative versions of Friedman's theorem. In the process, we study subgraph counters, compute rational formulas for the spectral statistics of adjacency matrices, and bound the error of asymptotic expansions for these spectral statistics.