Discussions about different graph Laplacians—mainly the normalized and unnormalized versions of graph Laplacian—have been ardent with respect to various methods of clustering and graph based semi-supervised learning. Previous research in the graph Laplacians, from a continuous perspective, investigated the convergence properties of the Laplacian operators on Riemannian Manifolds. In this paper,...

The question of what happens to the eigenvalues of the Laplacian of a graph when we delete a vertex is addressed. It is shown that λi − 1 ≤ λi ≤ λi+1, where λi is the ith smallest eigenvalues of the Laplacian of the original graph and λ v i is the ith smallest eigenvalues of the Laplacian of the graph G[V −v]; i.e., the graph obtained after removing the vertex v. It is shown that the average nu...

The question of what happens to the eigenvalues of the Laplacian of a graph when we delete a vertex is addressed. It is shown that λi − 1 ≤ λi ≤ λi+1, where λi is the ith smallest eigenvalues of the Laplacian of the original graph and λ v i is the ith smallest eigenvalues of the Laplacian of the graph G[V −v]; i.e., the graph obtained after removing the vertex v. It is shown that the average nu...

This is the third part of our work with a common title. The first [11] and the second part [12] will be also referred in the sequel as Part I and Part II, respectively. This third part was not planned at the beginning, but a lot of recently published papers on the signless Laplacian eigenvalues of graphs and some observations of ours justify its preparation. By a spectral graph theory we unders...

Contents Chapter 1. Eigenvalues and the Laplacian of a graph 1 1.1. Introduction 1 1.2. The Laplacian and eigenvalues 2 1.3. Basic facts about the spectrum of a graph 6

In this paper we give a simple characterization of the Laplacian spectra of a family of graphs as the eigenvalues of symmetric tridiagonal matrices. In addition, we apply our result to obtain an upper and lower bounds for the Laplacian-energy-like invariant of these graphs. The class of graphs considered are obtained by copies of modified generalized Bethe trees (obtained by joining the vertice...

Given a set of n randomly drawn sample points, spectral clustering in its simplest form uses the second eigenvector of the graph Laplacian matrix, constructed on the similarity graph between the sample points, to obtain a partition of the sample. We are interested in the question how spectral clustering behaves for growing sample size n. In case one uses the normalized graph Laplacian, we show ...

We define the Laplacian for a general graph and then examine several isoperimetric inequalities which relate the eigenvalues of the Laplacian to a number of graphs invariants such as vertex or edge expansions and the isoperimetric dimension of a graph.

A lower bound is given for the harmonic mean of the growth in a finite undirected graph 1 in terms of the eigenvalues of the Laplacian of 1. For a connected graph, this bound is tight if and only if the graph is distance-regular. Bounds on the diameter of a ``sphere-regular'' graph follow. Finally, a lower bound is given for the growth in an infinite undirected graph of bounded degree in terms ...