Related papers: Eigenvalues of the normalized complex Laplacian on…
The Laplacian spread of a graph is the difference between the largest eigenvalue and the second-smallest eigenvalue of the Laplacian matrix of the graph. We find that the class of strongly regular graphs attains the maximum of largest…
On compact Riemann surfaces, the Laplacian has a discrete, non-negative spectrum of eigenvalues of finite multiplicity. The spectrum is intrinsically linked to the geometry of the surface. In this work, we consider surfaces of constant…
Operators on unbounded domains may acquire eigenvalues that are embedded in the essential spectrum. Determining the fate of these embedded eigenvalues under small perturbations of the underlying operator is a challenging task, and the…
On a finite connected metric graph, we establish upper bounds for the eigenvalues of the Laplacian. These bounds depend on the length, the Betti number, and the number of pendant vertices. For trees, these estimates are sharp. We also…
The spectrum of the d-bar-Neumann Laplacian on a polydisc in several complex variables is explicitly computed. The calculation exhibits that the spectrum consists of eigenvalues, some of which, in particular the smallest ones, are of…
The study of complex networks has been one of the most active fields in science in recent decades. Spectral properties of networks (or graphs that represent them) are of fundamental importance. Researchers have been investigating these…
This paper is devoted to providing quantitative bounds on the location of eigenvalues, both discrete and embedded, of non self-adjoint Lam\'e operators of elasticity $-\Delta^\ast + V$ in terms of suitable norms of the potential $V$. In…
To some extent, graph evolutionary mechanisms can be explained by its spectra. Here, we are interested in two graph operations, namely, motif (subgraph) doubling and attachment that are biologically relevant. We investigate how these two…
In this paper, we characterize all connected graphs with exactly three distinct normalized Laplacian eigenvalues of which one is equal to $1$, determine all connected bipartite graphs with at least one vertex of degree $1$ having exactly…
The main objective of the present work is to study the negative spectrum of (differential) Laplace operators on metric graphs as well as their resolvents and associated heat semigroups. We prove an upper bound on the number of negative…
We consider two eigenvalue problems for Laplacian on some specific doubly connected domain. In particular, we study the following two eigenvalue problems. Let $B_1$ be an open ball in $\mathbb{R}^n$ and $B_0$ be a ball contained in $B_1$.…
The work in this thesis concerns the investigation of eigenvalues of the Laplacian matrix, normalized Laplacian matrix, signless Laplacian matrix and distance signless Laplacian matrix of graphs. In Chapter 1, we present a brief…
We study the eigenspace of the Laplacian matrix of a simple graph corresponding to the largest eigenvalue, subsequently arriving at the theory of modular decomposition of T. Gallai.
For $\alpha\in(0,\pi)$, let $U_\alpha$ denote the infinite planar sector of opening $2\alpha$, \[ U_\alpha=\big\{ (x_1,x_2)\in\mathbb R^2: \big|\arg(x_1+ix_2) \big|<\alpha \big\}, \] and $T^\gamma_\alpha$ be the Laplacian in…
We analyze the spectrum of the Laplace operator, subject to homogeneous complex magnetic fields in the plane. For real magnetic fields, it is well-known that the spectrum consists of isolated eigenvalues of infinite multiplicities (Landau…
In this paper we generalise the results on eigenvalues and eigenvectors of unnormalized (combinatorial) Laplacian of two-dimensional grid presented by Edwards:2013 first to a grid graph of any dimension, and second also to other types of…
We obtain geometric estimates for the first eigenvalue and the fundamental tone of the p-laplacian on manifolds in terms of admissible vector fields. Also, we defined a new spectral invariant and we show its relation with the geometry of…
Building on previous work that provided analytical solutions to generalised matrix eigenvalue problems arising from numerical discretisations, this paper develops exact eigenvalues and eigenvectors for a broader class of $n$-dimensional…
For the Erd\H{o}s-R\'enyi graph of size $N$ with mean degree $(1+o(1))\frac{\log N}{t+1}\leq d\leq(1-o(1))\frac{\log N}{t}$ where $t\in\mathbb{N}^{*}$, with high probability the smallest non zero eigenvalue of the Laplacian is equal to…
In the present paper we study some kinds of the problems for the bi-drifting Laplacian operator and get some sharp lower bounds for the first eigenvalue for these eigenvalue problems on compact manifolds with boundary (also called a smooth…