On the Normalized Laplacian Spectrum of Graphs and Graph Operations

For any graph Γ, we can define several matrices using the connection between its vertices. Evaluation of structural information of a graph from the eigenvalues of a matrix associated to it is the main goal of spectral graph theory. The adjacency matrix, Laplacian matrix, signless Laplacian matrix and normalized Laplacian matrix are the most popular matrices studied in spectral graph theory. In this thesis, our main focus is to analyze the eigenvalues of the normalized Laplacian matrix of a graph. For any undirected graph Γ (without any isolated vertices), let A be the (0,1)-adjacency matrix of Γ. The normalized Laplacian of Γ is Δ = I −D−1A, where D is the diagonal matrix of vertex degrees of Γ. If Γ is connected, then the matrix A = D⁻¹A is an irreducible row-stochastic matrix. The previously existing eigenvalue localization theorems did not provide satisfactory eigenvalue bounds for A. Here we come up with a new eigenvalue localization theorem, by applying which we get improved bounds for the eigenvalues of A as well as of Δ. The Randi´c index of a graph is useful to characterize that graph. It is also an useful tool to bound the normalized Laplacian energy of a graph. We generalize the concept of the Randi´c index and introduce some new topological indices. We call them general Randi´c indices for matching. We show that the coefficients of the characteristic polynomial, of the normalized Laplacian, of a tree can be expressed by these indices. Finally, we characterize these results for two special class of trees. It is not always easy to find the spectrum of a graph with large number of vertices and it is also seen that many real-world networks possess special type of graph structure. Creation of those structures can be explained by some graph operations, namely, vertex doubling, motif (induced subgraph) doubling, motif joining, etc. These graph operations produce certain eigenvalues, like 1, 1±0.5, 1±√0.5 etc, which are mostly observed in the normalized graph Laplacian of many real networks. For example, the doubling of a vertex always ensures the eigenvalue 1. We investigate the emergence of particular eigenvalues, such as, eigenvalue 1 and others by the above-mentioned graph operations. A threshold graph is an iterated graph. Production of a threshold graph can also be considered as a sequence of graph operations, starting from a single vertex, by repeatedly performing one of the two graph operations, namely, (a) addition of a single isolated vertex to the graph or (b) addition of a single dominating vertex to the graph. Thus a threshold graph can always be represented by a unique binary string starting with 0. We show that the unique string provides certain eigenvalues of that graph. Finally, we try to characterize threshold graphs with few distinct eigenvalues.