We prove that among all triangles of given diameter, the equilateral triangle minimizes the sum of the first n eigenvalues of the Neumann Laplacian, when n 3 . The result fails for n = 2 , because the second eigenvalue is known to be minimal for the degenerate acute isosceles triangle (rather than for the equilateral) while the first eigenvalue is 0 for every triangle. We show the third eigenva...

In this paper, a rigorous convergence and error analysis of a Galerkin boundary element method for the Dirichlet Laplacian eigenvalue problem is presented. The formulation of the eigenvalue problem in terms of a boundary integral equation yields a nonlinear boundary integral operator eigenvalue problem. This nonlinear eigenvalue problem and its Galerkin approximation are analyzed in the framewo...

We consider the problem of minimising the kth eigenvalue, k ≥ 2, of the (p-)Laplacian with Robin boundary conditions with respect to all domains in R of given volume M . When k = 2, we prove that the second eigenvalue of the p-Laplacian is minimised by the domain consisting of the disjoint union of two balls of equal volume, and that this is the unique domain with this property. For p = 2 and k...

Let G be a nonsingular connected mixed graph. We determine the mixed graphs G on at least seven vertices with exactly two Laplacian eigenvalues greater than 2. In addition, all mixed graphs G with exactly one Laplacian eigenvalue greater than 2 are also characterized. c © 2006 Elsevier Ltd. All rights reserved.

We consider Γ = (X, E) a dual polar graph and we give a tight frame on each eigenspace of the Laplacian operator associated to Γ. We compute the constants associated to each tight frame and as an application we give a formula for the product in the Norton algebra attached to the eigenspace corresponding to the second largest eigenvalue of the Laplacian.

In an important paper, Alon [2] derived a Cheeger–type inequality [8], by bounding from below the second smallest eigenvalue of the Laplacian of a finite undirected graph by a function of a (vertex) isoperimetric constant. More precisely, let G=(V,E) be a finite, undirected, connected graph, and let λ2(G) denote twice (for reasons explained below) the smallest non-zero eigenvalue of the Laplaci...

Let π : Z → Y be a Riemannian V -submersion of compact V manifolds. We study when the pull-back of an eigenform of the Laplacian on Y is an eigenform of the Laplacian on Z, and when the associated eigenvalue can change.

We give a method to construct cospectral graphs for the normalized Laplacian by swapping edges between vertices in some special graphs. We also give a method to construct an arbitrarily large family of (non-bipartite) graphs which are mutually cospectral for the normalized Laplacian matrix of a graph. AMS 2010 subject classification: 05C50

Cheeger’s fundamental inequality states that any edge-weighted graph has a vertex subset S such that its expansion (a.k.a. conductance of S or the sparsity of the cut (S, S̄)) is bounded as follows: φ(S) def = w(S, S̄) min{w(S), w(S̄)} 6 √ 2λ2, where w is the total edge weight of a subset or a cut and λ2 is the second smallest eigenvalue of the normalized Laplacian of the graph. We study three nat...

We prove an asymptotic estimate for the growth of variational eigenvalues of fractional p-Laplacian eigenvalue problems on a smooth bounded domain.