Let $W$ be an $n \times n$ symmetric matrix with i.i.d. centered entries with variance one. The celebrated Wigner’s theorem states that the empirical law of eigenvalues of $W$ converges weakly to the semicircular law, which is supported on $[-2,2]$. In addition, Bai and Yin proved that the largest eigenvalue of $W$ converges to 2 almost surely. In this case, we say that there are no outliers. In this talk, we explore the universality and stability of the Bai-Yin result. If we sparsify $W$ according to a regular graph $G$, can we find conditions under which there are no outliers? This is a joint work with Dylan Altschuler, Konstantin Tikhomirov, and Pierre Youssef.