We give sufficient conditions for existence of a perfect matching in a graph in terms of the eigenvalues of the Laplacian matrix. We also show that a distance-regular graph of degree k is k-edge-connected

We give some necessary conditions for a graph to be 3-chromatic in terms of the spectrum of the adjacency matrix. For all known distance-regular graphs it is determined whether they are 3-chromatic. A start is made with the classification of 3-chromatic distance-regular graphs, and it is shown...