Georgia Institute of Technology College of Sciences

The Laplacian of a graph is a real symmetric matrix given by $L=D-A$, where $D$ is the degree matrix of the graph and $A$ is the adjacency matrix. The Laplacian is a central object in spectral graph theory, and the spectrum of $L$ contains information on the graph. In the case of a random graph the Laplacian will be a random real symmetric matrix with dependent entries. These random Laplacian matrices can be generalized by taking $A$ to be a random real symmetric matrix and $D$ to be a diagonal matrix with entries equal to the row sums of $A$. We will consider the eigenvalues of general random Laplacian matrices, and the challenges raised by the dependence between $D$ and $A$. After discussing the bulk global eigenvalue behavior of general random Laplacian matrices, we will focus in detail on fluctuations of the largest eigenvalue of $L$ when $A$ is a matrix of independent Gaussian random variables. The asymptotic behavior of these Gaussian Laplacian matrices has a particularly nice free probabilistic interpretation, which can be exploited in the study of their eigenvalues. We will see how this interpretation can locate the largest eigenvalue of $L$ with respect to the largest entry of $D$. This talk is based on joint work with Kyle Luh and Sean O'Rourke.