We show that the action of conformal vector fields on functions on the sphere determines the spectrum of the Laplacian (or the conformal Laplacian), without further input of information. The spectra of intertwining operators (both differential and non-local) with principal part a power of the Laplacian follows as a corollary. An application of the method is the sharp form of Gross’ entropy ineq...

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

Abstract. Given two graphs G1, with vertices 1, 2, ..., n and edges e1, e2, ..., em, and G2, the edge corona G1 ⋄G2 of G1 and G2 is defined as the graph obtained by taking m copies of G2 and for each edge ek = ij of G, joining edges between the two end-vertices i, j of ek and each vertex of the k-copy of G2. In this paper, the adjacency spectrum and Laplacian spectrum of G1 ⋄ G2 are given in te...

In their study of the spectrum of quantum layers [6], Duclos, Exner, and Krejčǐŕık proved the existence of bound states for certain quantum layers . Part of their motivations to study the quantum layers is from mesoscopic physics. From the mathematical point of view, a quantum layer is a noncompact noncomplete manifold. For such a manifold, the spectrum of the Laplacian (with Dirichlet or Neuma...

We introduce a family of post-critically finite fractal trees indexed by the number of branches they possess. Then we produce a Laplacian operator on graph approximations to these fractals and use spectral decimation to describe the spectrum of the Laplacian on these trees. Lastly we consider the behavior of the spectrum as the number of branches increases. MCS: 28A80, 34B45, 15A18, 60J45, 94C9...

In spectral graph theory, the Grone-Merris Conjecture asserts that the spectrum of the Laplacian matrix of a finite graph is majorized by the conjugate degree sequence of this graph. We give a complete proof for this conjecture. The Laplacian of a simple graph G with n vertices is a positive semi-definite n×n matrix L(G) that mimics the geometric Laplacian of a Riemannian manifold; see §1 for d...

A graph is said to be determined by the adjacency and Laplacian spectrum (or to be a DS graph, for short) if there is no other non-isomorphic graph with the same adjacency and Laplacian spectrum, respectively. It is known that connected graphs of index less than 2 are determined by their adjacency spectrum. In this paper, we focus on the problem of characterization of DS graphs of index less th...

This is the first in a series of papers [Z3, Z4] on inverse spectral/resonance problems for analytic plane domains Ω. In this paper, we present a rigorous version of the Balian-Bloch trace formula [BB1, BB2]. It is an asymptotic formula for the trace Tr1ΩRρ(k + iτ log k) of the regularized resolvent of the Dirichlet Laplacian of Ω as k → ∞ with τ > 0. When the support of ρ̂ contains the length L...

This paper is devoted to the spectral analysis of the Laplacian with constant magnetic field on a cone of aperture α and Neumann boundary condition. We analyze the influence of the orientation of the magnetic field. In particular, for any orientation of the magnetic field, we prove the existence of discrete spectrum below the essential spectrum in the limit α → 0 and establish a full asymptotic...