We construct a determinant of the Laplacian for infinite-area surfaces which are hyperbolic near infinity and without cusps. In the case of a convex co-compact hyperbolic metric, the determinant can be related to the Selberg zeta function and thus shown to be an entire function of order two with zeros at the eigenvalues and resonances of the Laplacian. In the hyperbolic near infinity case the d...

Let G be a graph and L(G) be the Laplacian matrix of G. In this article, we first point out that the sequence of the moduli of Laplacian coefficients of G is log-concave and hence unimodal. Using this fact, we provide an upper bound for the partial sums of the Laplacian eigenvalues of G, based on coefficients of its Laplacian characteristic polynomial. We then obtain some lower bounds on the al...

We study the spectral properties of the Laplacian matrices and the normalized Laplacian matrices of the Erdös-Rényi random graph G(n, pn) for large n. Although the graph is simple, we discover some interesting behaviors of the two Laplacian matrices. In fact, under the dilute case, that is, pn ∈ (0, 1) and npn → ∞, we prove that the empirical distribution of the eigenvalues of the Laplacian mat...

We consider the method of particular solutions for numerically computing eigenvalues and eigenfunctions of the Laplacian on a smooth, bounded domain Ω in Rn with either Dirichlet or Neumann boundary conditions. This method constructs approximate eigenvalues E, and approximate eigenfunctions u that satisfy ∆u = Eu in Ω, but not the exact boundary condition. An inclusion bound is then an estimate...

In Lecture 10, we introduced a fundamental object of spectral graph theory: the graph Laplacian, and established some of its basic properties. We then focused on the task of estimating the value of eigenvalues of Laplacians. In particular, we proved the Courant-Fisher theorem that is instrumental in obtaining upper-bounding estimates on eigenvalues. Today, we continue by showing a technique – s...

In this paper we numerically solve the eigenvalue problem ∆u+λu = 0 on the fractal region defined by the Koch Snowflake, with zero-Dirichlet or zero-Neumann boundary conditions. The Laplacian with boundary conditions is approximated by a large symmetric matrix. The eigenvalues and eigenvectors of this matrix are computed by ARPACK. We impose the boundary conditions in a way that gives improved ...

, where deg(vi) is the sum of weights of all edges connected to vi. The signless Laplacian matrix Q(G) is defined by D(G) + A(G). We denote by 0 = λ1(G) ≤ λ2(G) ≤ · · · ≤ λn(G) the eigenvalues of L(G), and by μ1(G) ≤ μ2(G) ≤ · · · ≤ μn(G) the eigenvalues of Q(G). We order the degrees of the vertices of G as d1(G) ≤ d2(G) ≤ · · · ≤ dn(G). Various bounds for the Laplacian eigenvalues of unweighte...

It is well known that the main difficulty in solving eigenvalue problems under shape deformation relates to the continuation of multiple eigenvalues of the unperturbed configuration. These eigenvalues may evolve, under shape deformation, as separated, distinct eigenvalues. In this paper, we address the integral equation method in the evaluation of eigenfunctions and the corresponding eigenvalue...

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...