Explore recent advancements in random matrix theory through this comprehensive lecture. Delve into the spectral statistics of normalized adjacency matrices for random d-regular graphs on N vertices. Examine the eigenvalue rigidity and characteristic fluctuations for the Tracy-Widom law, with a focus on the second largest eigenvalue. Investigate the behavior of extremal eigenvalues for specific graph conditions. Gain insights into the joint work of Hong Tzer Yau, Jiaoyang Huang, and Theo McKenzie, which presents groundbreaking results in this field of mathematics.