Normalized Laplacian spectrum of graphs

In this project, we will use some modern techniques and methods of algebraic graph theory, linear algebra, matrix theory, combinatorics, and the software “Newgraph” to investigate the properties of the normalized Laplacian eigenvalues and give some sharp upper (or lower) bounds for the normalized Laplacian eigenvalues of graphs. We will also study the relationships between the normalized Laplacian eigenvalues of a graph and its invariants, how to use the normalized Laplacian eigenvalues to find properties of the graph, and make a connection between the adjacency matrix, the Laplacian and the normalized Laplacian matrix and their corresponding eigenvalues.