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.

We prove the “hot spots” conjecture of J. Rauch in the case that the domain Ω is a planar convex domain satisfying diam(Ω)2/|Ω| < 1.378. Specifically, we show that an eigenfunction corresponding to the lowest nonzero eigenvalue of the Neumann Laplacian on Ω attains its maximum (minimum) at points on ∂Ω. When Ω is a disk, diam(Ω)2/|Ω| t 1.273. Hence, the above condition indicates that Ω is a nea...

Abstract. The largest eigenvalue of a matrix is always larger or equal than its largest diagonal entry. We show that for a large class of random Laplacian matrices, this bound is essentially tight: the largest eigenvalue is, up to lower order terms, often the size of the largest diagonal entry. Besides being a simple tool to obtain precise estimates on the largest eigenvalue of a large class of...