This seminar talk from the "Spectral Geometry in the clouds" series features Theo McKenzie from Stanford University discussing precise eigenvalue location for random regular graphs. Explore the spectral theory of regular graphs and its applications in theoretical computer science, statistical physics, and mathematics. Learn about Ramanujan graphs, which have optimally large spectral gaps, and discover McKenzie's groundbreaking research showing that approximately 69% of randomly selected regular graphs are Ramanujan. Follow along as he presents a rigorous analysis of the Green's function of the adjacency operator, with particular focus on its behavior under random edge switches. This mathematical presentation was delivered on March 10, 2025, through the Centre de recherches mathématiques (CRM).