Related papers: Eigenvectors of the discrete Laplacian on regular …

For a regular polyhedron (or polygon) centered at the origin, the coordinates of the vertices are eigenvectors of the graph Laplacian for the skeleton of that polyhedron (or polygon) associated with the first (non-trivial) eigenvalue. In…

We analyze eigenvalues fluctuations of the Laplacian of various networks under the random matrix theory framework. Analyses of random networks, scale-free networks and small-world networks show that nearest neighbor spacing distribution of…

We introduce an unsupervised graph embedding that trades off local node similarity and connectivity, and global structure. The embedding is based on a generalized graph Laplacian, whose eigenvectors compactly capture both network structure…

The need to build a link between the structure of a complex network and the dynamical properties of the corresponding complex system (comprised of multiple low dimensional systems) has recently become apparent. Several attempts to tackle…

A metrized graph is a compact singular 1-manifold endowed with a metric. A given metrized graph can be modelled by a family of weighted combinatorial graphs. If one chooses a sequence of models from this family such that the vertices become…

Motivated by discrete Laplacian differential operators with various accuracy orders in numerical analysis, we introduce new matrices attached to a simple graph that can be considered graph Laplacians with higher accuracy. In particular, we…

Spectral methods that are based on eigenvectors and eigenvalues of discrete graph Laplacians, such as Diffusion Maps and Laplacian Eigenmaps are often used for manifold learning and non-linear dimensionality reduction. It was previously…

Let $d\geq 3$ be fixed and $G$ be a large random $d$-regular graph on $n$ vertices. We show that if $n$ is large enough then the entry distribution of every almost eigenvector $v$ of $G$ (with entry sum 0 and normalized to have length…

Spectra of sparse non-Hermitian random matrices determine the dynamics of complex processes on graphs. Eigenvalue outliers in the spectrum are of particular interest, since they determine the stationary state and the stability of dynamical…

We compare the spectrum and the localisation properties of the eigenmodes of the Laplacian and the adjacency matrix of 2D random geometric graphs, using numerical diagonalization of these matrices for different system sizes and…

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 deals with the problem of detecting non-isotropic high-dimensional geometric structure in random graphs. Namely, we study a model of a random geometric graph in which vertices correspond to points generated randomly and…

The original contributions of this paper are twofold: a new understanding of the influence of noise on the eigenvectors of the graph Laplacian of a set of image patches, and an algorithm to estimate a denoised set of patches from a noisy…

A central issue in the study of polymer physics is to understand the relation between the geometrical properties of macromolecules and various dynamics, most of which are encoded in the Laplacian spectra of a related graph describing the…

In this paper, we analyze the eigenfunctions of the edge-based Laplacian on a graph and the relationship of these functions to random walks on the graph. We commence by discussing the set of eigenfunctions supported at the vertices, and…

We examine the adjacency matrices of three-regular graphs representing one-face maps. Numerical studies reveal that the limiting eigenvalue statistics of these matrices are the same as those of much larger, and more widely studied classes…

Euclidean random matrices arise in a wide range of physical systems where interactions are determined by spatial configurations, including disordered media and cooperative phenomena in atomic ensembles. Unlike classical random matrix…

Graph neural networks (GNNs) have achieved remarkable success in a variety of machine learning tasks over graph data. Existing GNNs usually rely on message passing, i.e., computing node representations by gathering information from the…

We observe that the Laplacian of a random graph G on N vertices represents and explicitly solvable model in the limit of infinitely increasing N. Namely, we derive recurrent relations for the limiting averaged moments of the adjacency…

For a simple graph on $n$ vertices, any of its signless Laplacian eigenvalues is in the interval $[0, 2n-2]$. In this paper, we give relationships between the number of signless Laplacian eigenvalues in specific intervals in $[0, 2n-2]$ and…