For an n × 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/ √ 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.

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