The extreme eigenvalues of a random matrix have been important objects of study since the inception of random matrix theory and also have a variety of applications.  The Laplacian matrix is the workhorse of spectral graph theory and is the key player in many practical algorithms for graph clustering, network control theory and combinatorial optimization.  In this talk, we discuss the fluctuations of the extreme eigenvalues of a random Laplacian matrix with gaussian entries.  The proof relies on a broad set of techniques from random matrix theory and free probability.  We will also describe some recent progress on a broader class of random Laplacian matrices.

This is joint work with Andrew Campbell and Sean O'Rourke.