Related papers: Flexibility of eigenvalues for graph Laplacians ar…

We give inequalities relating the eigenvalues of the adjacency matrix and the Laplacian of a graph, and its minimum and maximum degrees. The results are applied to derive new conditions for quasi-randomness of graphs.

In this paper we study two classes of graphs, the (m,k)-stars and l-dependent graphs, investigating the relation between spectrum characteristics and graph structure: conditions on the topology and edge weights are given in order to get…

In this paper, we obtain some comparisons of the Dirichlet, Neumann and Laplacian eigenvalues on graphs. We also discuss their rigidities and some of their applications including some Lichnerowicz-type, Fiedler-type and Friedman-type…

This note introduces a result on the location of eigenvalues, i.e., the spectrum, of the Laplacian for a family of undirected graphs with self-loops. We extend on the known results for the spectrum of undirected graphs without self-loops or…

This paper establishes quantitative high-probability bounds on the eigenvalues and eigenfunctions of $\epsilon$-neighborhood graph Laplacians constructed from i.i.d. random variables on $m$-dimensional closed Riemannian manifolds $(M,g)$…

In this note we present upper bounds for the variational eigenvalues of the $p$-Laplacian on smooth domains of complete $n$-dimensional Riemannian manifolds and Neumann boundary conditions, and on compact (boundaryless) Riemannian…

We present a method for proving upper bounds on the eigenvalues of the graph Laplacian. A main step involves choosing an appropriate "Riemannian" metric to uniformize the geometry of the graph. In many interesting cases, the existence of…

We present sharp inequalities relating the number of vertices, edges, and triangles of a graph to the smallest eigenvalue of its adjacency matrix and the largest eigenvalue of its Laplacian.

This paper studies the Laplacian spectrum and the average effective resistance of (large) graphs that are sampled from graphons. Broadly speaking, our main finding is that the Laplacian eigenvalues of a large dense graph can be effectively…

In this paper, we successfully establish a Courant-type nodal domain theorem for both the Dirichlet eigenvalue problem and the closed eigenvalue problem of the Witten-Laplacian. Moreover, we also characterize the properties of the nodal…

We consider Laplacians on periodic equilateral metric graphs. The spectrum of the Laplacian consists of an absolutely continuous part (which is a union of an infinite number of non-degenerated spectral bands) plus an infinite number of flat…

We study the graphs associated with Vicsek sets in higher dimensional settings. First, we study the eigenvalues of the Laplacians on the approximating graphs of the Vicsek sets, finding a general spectral decimation function. This is an…

Let $G$ be a connected simple graph on $n$ vertices. Let $\mathcal{L}(G)$ be the normalized Laplacian matrix of $G$ and $\rho_{n-1}(G)$ be the second least eigenvalue of $\mathcal{L}(G)$. Denote by $\nu(G)$ the independence number of $G$.…

The famous Yang-Yau inequality provides an upper bound for the first eigenvalue of the Laplacian on an orientable Riemannian surface solely in terms of its genus $\gamma$ and the area. Its proof relies on the existence of holomorhic maps to…

In the present paper several bounds on multiplicities of eigenvalues of the Laplacian operator on surfaces are generalized from the case of either closed surface or simply-connected planar domain to the case of a surface of positive genus…

We develop a Hellmann--Feynman type perturbation theory for the discrete signed $p$-Laplacian and apply it to a parametrized perturbation by edge cuts. We show that the eigenvalues of the signed $p$-Laplacian can be characterized as citical…

In this article we classify compact Riemann surfaces of genus $1+q^2$ with a group of automorphisms of order $3q^2,$ where $q$ is a prime number. We also study decompositions of the corresponding Jacobian varieties.

Consider a semigraph $G=(V,\,E)$; in this paper, we study the eigenvalues of the Laplacian matrix of $G$. We show that the Laplacian of $G$ is positive semi-definite, and $G$ is connected if and only if $\lambda_2 >0.$ Along the similar…

We prove that eigenfunctions of the Laplacian on a compact hyperbolic surface delocalise in terms of a geometric parameter dependent upon the number of short closed geodesics on the surface. In particular, we show that an $L^2$ normalised…

The discrete Laplacian on Euclidean triangulated surfaces is a well-established notion. We introduce discrete Laplacians on spherical and hyperbolic triangulated surfaces. On the one hand, our definitions are close to the Euclidean one in…