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

We derive differential inequalities and difference inequalities for Riesz means of eigenvalues of the Dirichlet Laplacian, Rσ(z) := ∑ k (z − λk) σ +. Here {λk} ∞ k=1 are the ordered eigenvalues of the Laplacian on a bounded domain Ω ⊂ Rd, and x+ := max(0, x) denotes the positive part of the quantity x. As corollaries of these inequalities, we derive Weyl-type bounds on λk, on averages such as λ...

This paper focuses on an important aspect of cardiac surgical simulation, which is the deformation of mesh models to form smooth joins between them. A novel algorithm based on the Laplacian deformation method is developed. It extends the Laplacian method to handle deformation of 2-manifold mesh models with 1-D boundaries, and joining of 1-D boundaries to form smooth joins. Test results show tha...

The Mergelyan and Ahlfors-Beurling estimates for the Cauchy transform give quantitative information on uniform approximation by rational functions with poles off K. We will present an analogous result for an integral transform on the unit sphere in C2 introduced by Henkin, and show how it can be used to study approximation by functions that are locally harmonic with respect to the Kohn Laplacia...

We use a characterization of the fractional Laplacian as a Dirichlet to Neumann operator for an appropriate differential equation to study its obstacle problem in perforated domains.

In this paper, all connected bipartite graphs are characterized whose third largest Laplacian eigenvalue is less than three. Moreover, the result is used to characterize all connected bipartite graphs with exactly two Laplacian eigenvalues not less than three, and all connected line graphs of bipartite graphs with the third eigenvalue of their adjacency matrices less than one. c © 2003 Elsevier...

The spectrum of the ∂-Neumann Laplacian on a polydisc in C is explicitly computed. The calculation exhibits that the spectrum consists of eigenvalues, some of which, in particular the smallest ones, are of infinite multiplicity.

The purpose of this paper is to obtain a bound for sums of Hecke series in short intervals which, as a by-product, gives a new bound for Hj( 1 2 ). We begin by stating briefly the necessary notation and some results involving the spectral theory of the non-Euclidean Laplacian. For a competent and extensive account of spectral theory the reader is referred to Y. Motohashi’s monograph [13]. Let {...

This article follows the previous works [HKN] by Helffer-KleinNier and [HeNi1] by Helffer-Nier about the metastability in reversible diffusion processes via a Witten complex approach. Again, exponentially small eigenvalues of some self-adjoint realization of ∆ f,h = −h∆ + |∇f(x)| − h∆f(x) , are considered as the small parameter h > 0 goes to 0. The function f is assumed to be a Morse function o...