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.