Laplacian Eigenvalues Research Articles

A Steklov version of the torsional rigidity

Motivated by the connection between the first eigenvalue of the Dirichlet–Laplacian and the torsional rigidity, the aim of this paper is to find a physically coherent and mathematically interesting new concept for boundary torsional rigidity, closely related to the Steklov eigenvalue. From a variational point of view, such a new object corresponds to the sharp constant for the trace embedding of [Formula: see text] into [Formula: see text]. We obtain various equivalent variational formulations, present some properties of the state function and obtain some sharp geometric estimates, both for planar simply connected sets and for convex sets in any dimension.

Shape optimization for the Laplacian eigenvalue over triangles and its application to interpolation error analysis

A computer-assisted proof is proposed for the Laplacian eigenvalue minimization problems over triangular domains under diameter constraints. The paper provides an elementary analysis to derive the Hadamard shape derivative formula for complicated problem settings that allow non-homogeneous Neumann boundary conditions. Moreover, an algorithm to rigorously evaluate the Hadamard formula is proposed by utilizing the guaranteed computation for both eigenvalues and eigenfunctions of differential operators, which is recently developed by the second author. Besides the model homogeneous Dirichlet eigenvalue problem, the eigenvalue problem associated with a non-homogeneous Neumann boundary condition, which is related to the Crouzeix–Raviart interpolation error constant, is considered. The computer-assisted proof tells that among the triangles with the unit diameter, the equilateral triangle minimizes the first eigenvalue for each concerned eigenvalue problem.

FAST EXPANSION INTO HARMONICS ON THE DISK: A STEERABLE BASIS WITH FAST RADIAL CONVOLUTIONS.

We present a fast and numerically accurate method for expanding digitized images representing functions on [-1, 1]2 supported on the disk in the harmonics (Dirichlet Laplacian eigenfunctions) on the disk. Our method, which we refer to as the Fast Disk Harmonics Transform (FDHT), runs in operations. This basis is also known as the Fourier-Bessel basis, and it has several computational advantages: it is orthogonal, ordered by frequency, and steerable in the sense that images expanded in the basis can be rotated by applying a diagonal transform to the coefficients. Moreover, we show that convolution with radial functions can also be efficiently computed by applying a diagonal transform to the coefficients.

Geometric bounds for the magnetic Neumann eigenvalues in the plane

We consider the eigenvalues of the magnetic Laplacian on a bounded domain Ω of R2 with uniform magnetic field β>0 and magnetic Neumann boundary conditions. We find upper and lower bounds for the ground state energy λ1 and we provide semiclassical estimates in the spirit of Kröger for the first Riesz mean of the eigenvalues. We also discuss upper bounds for the first eigenvalue for non-constant magnetic fields β=β(x) on a simply connected domain in a Riemannian surface.In particular: we prove the upper bound λ1<β for a general plane domain for a constant magnetic field, and the upper bound λ1<maxx∈Ω‾⁡|β(x)| for a variable magnetic field when Ω is simply connected.For smooth domains, we prove a lower bound of λ1 depending only on the intensity of the magnetic field β and the rolling radius of the domain.The estimates on the Riesz mean imply an upper bound for the averages of the first k eigenvalues which is sharp when k→∞ and consists of the semiclassical limit 2πk|Ω| plus an oscillating term.We also construct several examples, showing the importance of the topology: in particular we show that an arbitrarily small tubular neighborhood of a generic simple closed curve has lowest eigenvalue bounded away from zero, contrary to the case of a simply connected domain of small area, for which λ1 is always small.

On the spectra of token graphs of cycles and other graphs

The k-token graph Fk(G) of a graph G is the graph whose vertices are the k-subsets of vertices from G, two of which being adjacent whenever their symmetric difference is a pair of adjacent vertices in G. It is a known result that the algebraic connectivity (or second Laplacian eigenvalue) of Fk(G) equals the algebraic connectivity of G. In this paper, we first give results that relate the algebraic connectivities of a token graph and the same graph after removing a vertex. Then, we prove the result on the algebraic connectivity of 2-token graphs for two infinite families: the odd graphs Or for all r, and the multipartite complete graphs Kn1,n2,…,nr for all n1,n2,…,nr In the case of cycles, we present a new method that allows us to compute the whole spectrum of F2(Cn). This method also allows us to obtain closed formulas that give asymptotically exact approximations for most of the eigenvalues of F2(Cn).

On Laplacian Eigenvalues of Wheel Graphs

Consider G to be a simple graph with n vertices and m edges, and L(G) to be a Laplacian matrix with Laplacian eigenvalues of μ1,μ2,…,μn=zero. Write Sk(G)=∑i=1kμi as the sum of the k-largest Laplacian eigenvalues of G, where k∈{1,2,…,n}. The motivation of this study is to solve a conjecture in algebraic graph theory for a special type of graph called a wheel graph. Brouwer’s conjecture states that Sk(G)≤m+k+12, where k=1,2,…,n. This paper proves Brouwer’s conjecture for wheel graphs. It also provides an upper bound for the sum of the largest Laplacian eigenvalues for the wheel graph Wn+1, which provides a better approximation for this upper bound using Brouwer’s conjecture and the Grone–Merris–Bai inequality. We study the symmetry of wheel graphs and recall an example of the symmetry group of Wn+1, n≥3. We obtain our results using majorization methods and illustrate our findings in tables, diagrams, and curves.

On the Laplacian spectrum of k-symmetric graphs

For some positive integer k, if the finite cyclic group Zk can freely act on a graph G, then we say that G is k-symmetric. In 1985, Faria showed that the multiplicity of Laplacian eigenvalue 1 is greater than or equal to the difference between the number of pendant vertices and the number of quasi-pendant vertices. But if a graph has a pendant vertex, then it is at most 1-connected. In this paper, we investigate a class of 2-connected k-symmetric graphs with a Laplacian eigenvalue 1. We also identify a class of k-symmetric graphs in which all Laplacian eigenvalues are integers.

An optimal lower bound in fractional spectral geometry for planar sets with topological constraints

AbstractWe prove a lower bound on the first eigenvalue of the fractional Dirichlet–Laplacian of order on planar open sets, in terms of their inradius and topology. The result is optimal, in many respects. In particular, we recover a classical result proved independently by Croke, Osserman, and Taylor, in the limit as goes to 1. The limit as goes to is carefully analyzed, as well.

Some new inequalities on the normalized signless Laplacian resolvent energy of graphs

For a connected graph [Formula: see text] of order [Formula: see text] with normalized signless Laplacian eigenvalues [Formula: see text], the normalized signless Laplacian resolvent energy of [Formula: see text] is defined as [Formula: see text]. In this paper, we derive new inequalities on [Formula: see text] as well as relations on [Formula: see text] with Randić (normalized) incidence energy. We also deduced that some of our results improve the existing results in the literature.

Proof of a conjecture on distribution of Laplacian eigenvalues and diameter, and beyond

Ahanjideh, Akbari, Fakharan and Trevisan proposed a conjecture on the distribution of the Laplacian eigenvalues of graphs: for any connected graph of order n with diameter d≥2 that is not a path, the number of Laplacian eigenvalues in the interval [n−d+2,n] is at most n−d. We show that the conjecture is true, and give a complete characterization of graphs for which the conjectured bound is attained. This establishes an interesting relation between the spectral and classical parameters.

Minimizations of positive periodic and Dirichlet eigenvalues for general indefinite Sturm-Liouville problems

The aim of this paper is to develop an analytical approach to obtain the sharp estimates for the lowest positive periodic eigenvalue and all Dirichlet eigenvalues of a general Sturm-Liouville problemy″=q(t)y+λm(t)y, where q is a nonnegative potential and another potential m admits to change sign. A typical example of such problems is the well-known Camassa-Holm equations with indefinite potentials, which corresponds to the case q(t)≡14. It is shown that the solution of the minimization problems of the lowest positive periodic eigenvalues and Dirichlet eigenvalues will lead to more general distributions of potentials which have no densities with respect to the Lebesgue measure. As a result, it is very natural to choose the general setting of the measure differential equationsdy•=y(t)dμ(t)+λydν(t), to understand the eigenvalues and their minimization, where μ and ν are two suitable measures. The variational characterization of lowest positive eigenvalues, together with a strong continuous dependence of eigenvalues on the potentials, will play crucial roles in our analysis. Different from the periodic case, we are able to obtain the optimal bounds for higher Dirichlet eigenvalues.

Classification of trees by Laplacian eigenvalue distribution and edge covering number

For a connected graph G of order n and an interval I, denote by mGI the number of Laplacian eigenvalues of G in I. In this paper, we present bounds for mGI in terms of the structural parameter β′(G), the edge covering number of G. We prove a known result that mG[1,n]≥β′(G). We also show that all graphs G≠C3,C7 with minimum degree at least two, mG[1,n]≥β′(G)+1. We present a short proof of the known result that mG(n−1,n]≤κ(G), where κ(G) is the vertex connectivity of G. Additionally, we classify all trees T such that mT(n−i,n]=j, for 1≤i, j≤2.

(Generalized) Incidence and Laplacian-Like Energies

In this study, for graph Γ with r connected components (also for connected nonbipartite and connected bipartite graphs) and a real number ε ≠ 0,1 , we found generalized and improved bounds for the sum of ε -th powers of Laplacian and signless Laplacian eigenvalues of Γ . Consequently, we also generalized and improved results on incidence energy I E and Laplacian energy-like invariant L E L .

Laplacian eigenvalues of the unit graph of the ring [formula omitted

We investigate the Laplacian eigenvalues about unit graph G(Zn) on Zn and show the case whenever n is an even number or an odd prime power, G(Zn) would be Laplacian integral. We also prove that if n>1, then the Laplacian spectral radius of G(Zn) is equal cardinal number of V(G(Zn)) if and only if n=pk, here p is a prime integer, k is an integer that positive. In addition, this paper also characterizes n that algebraic connectivity of G(Zn) coincides with the vertex connectivity.

Bounding the sum of the largest signless Laplacian eigenvalues of a graph

We show several sharp upper and lower bounds for the sum of the largest eigenvalues of the signless Laplacian matrix. These bounds improve and extend previously known bounds.

Nonsmooth Control Barrier Function design of continuous constraints for network connectivity maintenance

This paper considers the problem of maintaining global connectivity of a multi-robot system while executing a desired coordination task. Our approach builds on optimization-based feedback design formulations, where the nominal cost function and constraints encode desirable control objectives for the resulting input. Our solution uses the algebraic connectivity of the multi-robot interconnection topology as a control barrier function and critically embraces its nonsmooth nature. We take advantage of the understanding of how Laplacian eigenvalues behave as their multiplicities change, in combination with the flexibility provided by the concept of control barrier function, to carefully design additional constraints that guarantee the resulting optimization-based controller is continuous and maintains network connectivity. The technical treatment combines elements from set-valued theory, nonsmooth analysis, and algebraic graph theory to imbue the proposed constraints with regularity properties so that they can be smoothly combined with other control constraints. We provide simulations and experimental results illustrating the effectiveness and continuity of the proposed approach in a resource gathering problem.

On the spread of the distance signless Laplacian matrix of a graph

Abstract Let G be a connected graph with n vertices, m edges. The distance signless Laplacian matrix DQ(G) is defined as DQ(G) = Diag(Tr(G)) + D(G), where Diag(Tr(G)) is the diagonal matrix of vertex transmissions and D(G) is the distance matrix of G. The distance signless Laplacian eigenvalues of G are the eigenvalues of DQ(G) and are denoted by δ1 Q(G), δ2 Q(G), ..., δn Q(G). δ1 Q is called the distance signless Laplacian spectral radius of DQ(G). In this paper, we obtain upper and lower bounds for SDQ (G) in terms of the Wiener index, the transmission degree and the order of the graph.

Spectral estimates for the Dirichlet Laplacian on spiral-shaped regions

We derive spectral estimates of the Lieb–Thirring type for eigenvalues of Dirichlet Laplacians on strictly shrinking spiral-shaped domains.

Scaling inequalities for spherical and hyperbolic eigenvalues

Neumann and Dirichlet eigenvalues of the Laplacian on spherical and hyperbolic domains are shown to satisfy scaling inequalities or monotonicities analogous to the length ^{-2} scaling relation in Euclidean space. For a cap of aperture \Theta on the sphere \mathbb{S}^2 , normalizing the k -th eigenvalue by the square of the Euclidean radius of the boundary circle yields that \mu_k(\Theta) \sin^2 \Theta is strictly decreasing, while normalizing by the stereographic radius squared gives that \mu_k(\Theta) 4 \tan^2 \Theta/2 is strictly increasing. For the second Neumann eigenvalue, normalizing instead by the cap area establishes the stronger result that \mu_2(\Theta) 4 \sin^2 \Theta/2 is strictly increasing. Monotonicities of this kind are somewhat surprising, since the Neumann eigenvalues themselves can vary non-monotonically. Cheng and Bandle-type inequalities are deduced by assuming either fixed radius or fixed area and comparing eigenvalues of disks having different curvatures.

On the spectral gap in the Kac–Luttinger model and Bose–Einstein condensation

We consider the Dirichlet eigenvalues of the Laplacian among a Poissonian cloud of hard spherical obstacles of fixed radius in large boxes of Rd, d≥2. In a large box of side-length 2ℓ centered at the origin, the lowest eigenvalue is known to be typically of order (logℓ)−2/d. We show here that with probability arbitrarily close to 1 as ℓ goes to infinity, the spectral gap stays bigger than σ(logℓ)−(1+2/d), where the small positive number σ depends on how close to 1 one wishes the probability. Incidentally, the scale (logℓ)−(1+2/d) is expected to capture the correct size of the gap. Our result involves the proof of new deconcentration estimates. Combining this lower bound on the spectral gap with the results of Kerner–Pechmann–Spitzer, we infer a type-I generalized Bose–Einstein condensation in probability for a Kac–Luttinger system of non-interacting bosons among Poissonian spherical impurities, with the sole macroscopic occupation of the one-particle ground state when the density exceeds the critical value.