Laplacian Spectrum and Graph Structure

The Laplacian matrix of a graph and its eigenvalues can be used in several areas of mathematical research and have a physical interpretation in various physical and chemical theories. The related matrix the adjacency matrix of a graph and its eigenvalues were much more investigated in the past than the Laplacian matrix. However in my opinion the Laplacian spectrum is much more natural and more important than the adjacency matrix spectrum. We have used the standard terminology of graph theory as it is introduced in most text books on the theory of graphs. We have studied the graph structure related to the second smallest and the largest Laplacian eigenvalues.