In this paper, all connected graphs with the fourth largest Laplacian eigenvalue less than two are determined, which are used to characterize all connected graphs with exactly three Laplacian eigenvalues no less than two. Moreover, we determine bipartite graphs such that the adjacency matrices of their line graphs have exactly three nonnegative eigenvalues. © 2003 Elsevier Ltd. All rights reser...

In this paper, various modifications of a connected graph G are regarded as perturbations of its signless Laplacian matrix. Several results concerning the resulting changes to the signless Laplacian spectral radius of G are obtained by solving intermediate eigenvalue problems of the second type. AMS subject classifications: 05C50

Let (M, g) be a compact Riemannian manifold of dimension ≥ 3. We show that there is a metrics g̃ conformal to g and of volume 1 such that the first positive eigenvalue the conformal Laplacian with respect to g̃ is arbitrarily large. A similar statement is proven for the first positive eigenvalue of the Dirac operator on a spin manifold of dimension ≥ 2.

We discuss the behavior of the minimal eigenvalue λ of the Dirichlet Laplacian in the domainD1\D2 := D (an annulus) whereD1 is a circular disc andD2 ⊂ D1 is a smaller circular disc. It is conjectured that the minimal eigenvalue λ has a maximum value when D2 is a concentric disc. If h is a displacement of the center of the disc D2 and λ(h) is the corresponding minimal eigenvalue, then dλ(h) dh <...

We consider the Neumann Laplacian with constant magnetic field on a regular domain. Let B be the strength of the magnetic field, and let λ1(B) be the first eigenvalue of the magnetic Neumann Laplacian on the domain. It is proved that B 7→ λ1(B) is monotone increasing for large B. Combined with the results of [FoHe2], this implies that all the ‘third’ critical fields for strongly Type II superco...

The Yamabe invariant of a smooth compact manifold is by definition the supremum of the scalar curvatures of unit-volume Yamabe metrics on the manifold. For an explicit infinite class of 4-manifolds, we show that this invariant is positive but strictly less than that of the 4-sphere. This is done by using spin Dirac operators to control the lowest eigenvalue of a perturbation of the Yamabe Lapla...

In this paper, we investigate how the smallest signless Laplacian eigenvalue of a graph behaves when the graph is perturbed by deleting a vertex, subdividing edges or moving edges.

Let G be a connected simple graph whose Laplacian eigenvalues are 0 = μn (G) μn−1 (G) · · · μ1 (G) . In this paper, we establish some upper and lower bounds for the algebraic connectivity and the largest Laplacian eigenvalue of G . Mathematics subject classification (2010): 05C50, 15A18.

We analyze the limit of the p-form Laplacian under a collapse with bounded sectional curvature and bounded diameter to a singular limit space. As applications, we give results about upper and lower bounds on the j-th eigenvalue of the p-form Laplacian, in terms of sectional curvature and diameter.