**Title:** The Spectrum of the Connection Laplacian

**Abstract:** Connection graphs are weighted simple graphs in which every edge is associated with a scalar weight as well as a rotation matrix of some fixed dimension. Connection graphs arise in applications dealing with high dimensional data sets where a scalar weight alone is not sufficient to quantify the information on an edge. High dimensional versions of the adjacency matrix, the combinatorial Laplacian matrix, and the normalized Laplacian matrix can be defined for connection graphs. Just as in spectral graph theory for simple graphs, the eigenvalues of these matrices give us information about the structure of a connection graph. We examine some such properties, and give results from spectral graph theory generalized to connection graphs.