Related papers: Computing the $p$-Laplacian eigenpairs of signed g…
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…
The construction of a meaningful graph plays a crucial role in the success of many graph-based representations and algorithms for handling structured data, especially in the emerging field of graph signal processing. However, a meaningful…
Signed graphs are equipped with both positive and negative edge weights, encoding pairwise correlations as well as anti-correlations in data. A balanced signed graph has no cycles of odd number of negative edges. Laplacian of a balanced…
We establish a sharp lower bound on the first non-trivial eigenvalue of the Laplacian on a metric graph equipped with natural (i.e., continuity and Kirchhoff) vertex conditions in terms of the diameter and the total length of the graph.…
In this paper we consider particular graphs defined by $\overline{\overline{\overline{K_{\alpha_1}}\cup K_{\alpha_2}}\cup\cdots \cup K_{\alpha_k}}$, where $k$ is even, $K_\alpha$ is a complete graph on $\alpha$ vertices, $\cup$ stands for…
It has been recently shown that a large class of balanced graph cuts allows for an exact relaxation into a nonlinear eigenproblem. We review briefly some of these results and propose a family of algorithms to compute nonlinear eigenvectors…
We study the problem of estimating eigenpairs of elliptic differential operators from samples of a distribution $\rho$ supported on a manifold $M$. The operators discussed in the paper are relevant in unsupervised learning and in particular…
Let $G$ be a simple graph with the Laplacian matrix $L(G)$ and let $e(G)$ be the number of edges of $G$. A conjecture by Brouwer and a conjecture by Grone and Merris state that the sum of the $k$ largest Laplacian eigenvalues of $G$ is at…
The error estimation for eigenvalues and eigenvectors of a small positive symmetric perturbation on the spectrum of a graph Laplacian is related to Gau{\ss} hypergeometric functions. Based on this, a heuristic polynomial-time algorithm for…
Recently, Braunstein et al. [1] introduced normalized Laplacian matrices of graphs as density matrices in quantum mechanics and studied the relationships between quantum physical properties and graph theoretical properties of the underlying…
Given i.i.d. observations uniformly distributed on a closed manifold $\mathcal{M}\subseteq \mathbb{R}^p$, we study the spectral properties of the associated empirical graph Laplacian based on a Gaussian kernel. Our main results are…
We characterize all graphs for which there are eigenvectors of the graph Laplacian having all their components in {-1,+1} or {-1,0,+ 1}. Graphs having eigenvectors with components in {-1,+1} are called bivalent and are shown to be the…
In this paper, we assume that $q>0$, $p>1$ and $s\in(0,1)$ , and consider the following nonlinear fractional p-Laplacian equations on finite graphs: \begin{equation*} \left\{ \begin{array}{lll} \partial_t u^q(x,t)+(-\Delta)_p^su=0,\\[15pt]…
We establish metric graph counterparts of Pleijel's theorem on the asymptotics of the number of nodal domains $\nu_n$ of the $n$-th eigenfunction(s) of a broad class of operators on compact metric graphs, including Schr\"odinger operators…
We consider the normalized Laplace operator for directed graphs with positive and negative edge weights. This generalization of the normalized Laplace operator for undirected graphs is used to characterize directed acyclic graphs. Moreover,…
We introduce and study Laplacians on a finite metric graph endowed with generalized densities, that is, measures of finite mass. One important motivation is that this setting provides a common framework for several interesting classes of…
Several invariants of polarized metrized graphs and their applications in Arithmetic Geometry are studied recently. In this paper, we give fast algorithms to compute these invariants by expressing them in terms of the discrete Laplacian…
In this paper we are concerned with various graph invariants (girth, diameter, expansion constants, eigenvalues of the Laplacian, tree number) and their analogs for weighted graphs -- weighing the graph changes a combinatorial problem to…
Let $G$ be a graph with $n$ vertices. We denote the largest signless Laplacian eigenvalue of $G$ by $q_1(G)$ and Laplacian eigenvalues of $G$ by $\mu_1(G)\ge\cdots\ge\mu_{n-1}(G)\ge\mu_n(G)=0$. It is a conjecture on Laplacian spread of…
We prove explicit and sharp eigenvalue estimates for Neumann $p$-Laplace eigenvalues in domains that admit a representation in Fermi coordinates. More precisely, if $\gamma$ denotes a non-closed curve in $\mathbb{R}^2$ symmetric with…