Related papers: Sharp bounds for the largest eigenvalue

In this paper, we derive an upper bound for higher order eigenvalues of the normalized Laplace operator associated with a symmetric finite graph in terms of lower order eigenvalues.

For a graph with largest normalized Laplacian eigenvalue $\lambda_N$ and (vertex) coloring number $\chi$, it is known that $\lambda_N\geq \chi/(\chi-1)$. Here we prove properties of graphs for which this bound is sharp, and we study the…

We offer a new method for proving that the maximal eigenvalue of the normalized graph Laplacian of a graph with $n$ vertices is at least $\frac{n+1}{n-1}$ provided the graph is not complete and that equality is attained if and only if the…

For a given complex square matrix $A$ with constant row sum, we establish two new eigenvalue inclusion sets. Using these bounds, first we derive bounds for the second largest and smallest eigenvalues of adjacency matrices of $k$-regular…

Let $u$ be an eigenfunction of the Laplacian on a compact manifold with boundary, with Dirichlet or Neumann boundary conditions, and let $-\lambda^2$ be the corresponding eigenvalue. We consider the problem of estimating the maximum of $u$…

Let $H=-D^2+V$ be a Schr\"odinger operator on $ L^2(\mathbb{R})$, or on $ L^2(0,\infty)$. Suppose the potential satisfies $\limsup_{x\to \infty}|xV(x)|=a<\infty$. We prove that $H$ admits no eigenvalue larger than $ \frac{4a^2}{\pi^2}$. For…

In this paper, we give some lower bounds for several eigenvalues. Firstly, we investigate the eigenvalues $\lambda_i$ of the Laplace operator and prove a sharp lower bound. Moreover, we extent this estimate of the eigenvalues to general…

It is known that, for an oriented hypergraph with (vertex) coloring number $\chi$ and smallest and largest normalized Laplacian eigenvalues $\lambda_1$ and $\lambda_N$, respectively, the inequality $\chi\geq…

We prove asymptotically optimal upper bounds for the eigenvalues of the Wentzel-Laplace operator on Riemannian manifolds with Ricci curvature bounded below. These bounds depend highly on the geometry of the boundary in addition to the…

Let $M$ be a closed hypersurface in a noncompact rank-1 symmetric space $(\bar{\mathbb{M}}, ds^2)$ with $-4 \leq K_{\bar{\mathbb{M}}} \leq -1,$ or in a complete, simply connected Riemannian manifold $\mathbb{M}$ such that $0 \leq…

In this article we prove upper bounds for the Laplace eigenvalues $\lambda_k$ below the essential spectrum for strictly negatively curved Cartan-Hadamard manifolds. Our bound is given in terms of $k^2$ and specific geometric data of the…

In this paper, we consider the bounds for the largest eigenvalue and the sum of the $k$ largest Laplacian eigenvalues of signed graphs. Firstly, we give an upper bound on the largest eigenvalue of the adjacency matrix of a signed graph and…

We obtain sharp lower bounds for the first eigenvalue of four types of eigenvalue problem defined by the bi-Laplace operator on compact manifolds with boundary and determine all the eigenvalues and the corresponding eigenfunctions of a…

For perturbed Stark operators $Hu=-u^{\prime\prime}-xu+qu$, the author has proved that $\limsup_{x\to \infty}{x}^{\frac{1}{2}}|q(x)|$ must be larger than $\frac{1}{\sqrt{2}}N^{\frac{1}{2}}$ in order to create $N$ linearly independent…

In this article, we study sharp bounds for the Neumann eigenvalues of the Laplace operator on graphs. We first obtain monotonicity results for the Neumann eigenvalues on trees. In particular, we show that increasing any number of boundary…

The eigenvalue bounds obtained earlier [J. Phys. A: Math. Gen. 31 (1998) 963] for smooth transformations of the form V(x) = g(x^2) + f(1/x^2) are extended to N-dimensions. In particular a simple formula is derived which bounds the…

We derive quantitative bounds for eigenvalues of complex perturbations of the indefinite Laplacian on the real line. Our results substantially improve existing results even for real-valued potentials. For $L^1$-potentials, we obtain optimal…

Here we have investigated a few properties of the eigenvalues of normalized (geometric) graph Laplacian in different graphs. Preservation of eigenvalue 1 from a particular subgraph to the entire graph, the spectrum of the graph constructed…

We show that for any hyperbolic surface of genus g, the eigenvalue $\lambda _{2g-2}$ of the Laplace operator is > 1/4.

We introduce the discrete poly-Laplace operator on a subgraph with Dirichlet boundary condition. We obtain upper and lower bounds for the sum of the first $k$ Dirichlet eigenvalues of the poly-Laplace operators on a finite subgraph of…