  A Laplacian matrix is a real symmetric matrix whose row and column sums are
zero. We investigate the limiting distribution of the largest eigenvalue of a
Laplacian random matrix with Gaussian entries. Unlike many classical matrix
ensembles, this random matrix model contains dependent entries. After properly
shifting and scaling, we show the largest eigenvalue converges to the Gumbel
distribution as the dimension of the matrix tends to infinity. While the
largest diagonal entry is also shown to have Gumbel fluctuations, there is a
rather surprising difference between its deterministic centering term and the
centering term required for the largest eigenvalue.
