For eigenvalue problems of self-adjoint differential operators, a universal framework is proposed to give explicit lower and upper bounds for the eigenvalues. In the case of the Laplacian operator, by applying Crouzeix–Raviart finite elements, an efficient algorithm is developed to bound the eigenvalues for the Laplacian defined in 1D, 2D and 3D spaces. Moreover, for nonconvex domains, for whic...

Let O1 be a (cocompact) Fuchsian group, given as the group of units of norm one in a maximal order O in an indefinite quaternion division algebra over Q. Using the (classical) Selberg trace formula, we show that the eigenvalues of the automorphic Laplacian for O1 and their multiplicities coincide with the eigenvalues and multiplicities of the Laplacian defined on the Maaß newforms for the Hecke...

The aim of this paper is twofold: First, we characterize an essentially optimal class of boundary operators Θ which give rise to self-adjoint Laplacians −∆Θ,Ω in L (Ω; dx) with (nonlocal and local) Robin-type boundary conditions on bounded Lipschitz domains Ω ⊂ R, n ∈ N, n ≥ 2. Second, we extend Friedlander’s inequalities between Neumann and Dirichlet Laplacian eigenvalues to those between nonl...

Let O 1 be a (cocompact) Fuchsian group, given as the group of units of norm one in a maximal order O in an indeenite quaternion division algebra over Q. Using the (classical) Selberg trace formula, we show that the eigenvalues of the automorphic Laplacian for O 1 and their multiplicities coincide with the eigenvalues and multiplicities of the Laplacian deened on the Maaa newforms for the Hecke...

For a simple graph G, let e(G) denote the number of edges and Sk(G) denote the sum of the k largest eigenvalues of the signless Laplacian matrix of G. We conjecture that for any graph G with n vertices, Sk(G) ≤ e(G) + k+1 2 for k = 1, . . . , n. We prove the conjecture for k = 2 for any graph, and for all k for regular graphs. The conjecture is an analogous to a conjecture by A.E. Brouwer with ...

Series expansions are obtained for a rich subset of eigenvalues and eigenfunctions of an operator that arises in the study of rectangular membranes: the operator is the 2-D Laplacian with restorative force term and Dirichlet boundary conditions. Expansions are extracted by considering the restorative force term as a linear perturbation of the Laplacian; errors of truncation for these expansions...

This paper presents some bounds on the number of Laplacian eigenvalues contained in various subintervals of [0, n] by using the matching number and edge covering number for G, and asserts that for a connected graph the Laplacian eigenvalue 1 appears with certain multiplicity. Furthermore, as an application of our result (Theorem 13), Grone and Merris’ conjecture [The Laplacian spectrum of graph...

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...

Let G be a graph with n vertices and m edges. Let λ1, λ2, . . . , λn be the eigenvalues of the adjacency matrix of G, and let μ1, μ2, . . . , μn be the eigenvalues of the Laplacian matrix of G. An earlier much studied quantity E(G) = ∑ni=1 |λi | is the energy of the graph G. We now define and investigate the Laplacian energy as LE(G) = ∑ni=1 |μi − 2m/n|. There is a great deal of analogy between...

where n is the number of vertices of the graph G, and λ1,λ2, . . .,λn are its eigenvalues [1, 4, 5]. Two elementary properties of the graph energy are E(G1 ∪G2) = E(G1) + E(G2) for G1 ∪G2 being the graph consisting of two disconnected components G1 and G2, and E(G∪K1) = E(G), where K1 is the graph with a single vertex. Motivated by the success of the graph-energy concept, and in order to extend...