  For an $n\times n$ Laplacian random matrix $L$ with Gaussian entries it is
proven that the fluctuations of the largest eigenvalue and the largest diagonal
entry of $L/\sqrt{n-1}$ are Gumbel. We first establish suitable non-asymptotic
estimates and bounds for the largest eigenvalue of $L$ in terms of the largest
diagonal element of $L$. An expository review of existing results for the
asymptotic spectrum of a Laplacian random matrix is also presented, with the
goal of noting the differences from the corresponding classical results for
Wigner random matrices. Extensions to Laplacian block random matrices are
indicated.
