Laplacian Eigenvalues Research Articles

Spectrum of the Cozero-Divisor Graph Associated to Ring Zn

Let R be a commutative ring with identity 1≠0 and let Z(R)′ be the set of all non-unit and non-zero elements of ring R. Γ′(R) denotes the cozero-divisor graph of R and is an undirected graph with vertex set Z(R)′, w∉zR, and z∉wR if and only if two distinct vertices w and z are adjacent, where qR is the ideal generated by the element q in R. In this article, we investigate the signless Laplacian eigenvalues of the graphs Γ′(Zn). We also show that the cozero-divisor graph Γ′(Zp1p2) is a signless Laplacian integral.

Stability for an inverse spectral problem of the biharmonic Schrödinger operator

Let V be a bounded potential. We prove a Hölder stability estimate of determining the potential V from the boundary spectral data of the biharmonic operator Δ2+V. The boundary spectral data consists of the Dirichlet eigenvalues λk and the normal derivatives of the eigenfunctions ∂νϕk,∂νΔϕk on the boundary of a bounded domain. The analysis depends on the asymptotic behavior of the spectral data.

Computing Dirichlet eigenvalues of the Schrödinger operator with a PT-symmetric optical potential

We provide estimates for the eigenvalues of non-self-adjoint Sturm–Liouville operators with Dirichlet boundary conditions for a shift of the special potential 4cos ^{2}x+4iVsin 2x that is a PT-symmetric optical potential, especially when |c|=|sqrt{1-4V^{2}}|<2 or correspondingly 0leq V<sqrt {5}/2. We obtain some useful equations for calculating Dirichlet eigenvalues also for |c|geq 2 or equally Vgeq sqrt{5}/2. We discuss our results by comparing them with the periodic and antiperiodic eigenvalues of the Schrödinger operator. We even approximate complex eigenvalues by the roots of some polynomials derived from some iteration formulas. Moreover, we give a numerical example with error analysis.

On sum of powers of normalized Laplacian eigenvalues and resistance distances of graphs

Let G=(V,E) be a simple connected graph with vertex set V={1,2,…,n} and edge set E. The normalized Laplacian matrix of G is L=D−1/2(D−A)D−1/2, where D is the degree-diagonal matrix of G and A is the adjacency matrix of G. Let λ1≥λ2≥⋯≥λn−1≥λn=0 be the eigenvalues of L, for any real number α, the topological index sα∗(G) of G is defined as sα∗(G)=∑i=1n−1λiα.Let Ωij denote the resistance distance between vertices i and j, the degree-Kirchhoff index Kf∗(G) of G is defined as Kf∗(G)=∑i<jdidjΩij,where di is the degree of the vertex i. In this paper, we first express sα∗(G) in terms of Ωij’s explicitly for any real number α. Then as an application, we give a new proof for the classical Foster’s kth formula of resistance distances. Finally, we generalize the well-known relation 2|E|s−1∗(G)=Kf∗(G)to any integer k≥−1, which provides some upper bounds for sk∗(G).

Asymptotic behavior of generalized capacities with applications to eigenvalue perturbations: The higher dimensional case

We provide a full series expansion of a generalization of the so-called u-capacity related to the Dirichlet–Laplacian in dimension three and higher, extending the results of Abatangelo et al. (2021); Abatangelo, Léna and Musolino (2022) dealing with the planar case. We apply the result in order to study the asymptotic behavior of perturbed eigenvalues when Dirichlet conditions are imposed on a small regular subset of the domain of the eigenvalue problem.

On variational competing [formula omitted]Laplacian Dirichlet problem with gradient depending weight

We investigate the existence of variational generalized solutions to coercive competitive equations driven by a competing p,q−Laplacian type operator with weigth depending on the gradient. An abstract existence result based on a Galerkin scheme is therefore obtained and subsequently applied.

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.