Related papers: Spectral Convergence Rate of Graph Laplacian
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 propose two related unsupervised clustering algorithms which, for input, take data assumed to be sampled from a uniform distribution supported on a metric space $X$, and output a clustering of the data based on the selection of a…
Graph spectra have been successfully used to classify network types, compute the similarity between graphs, and determine the number of communities in a network. For large graphs, where an eigen-decomposition is infeasible, iterative moment…
We examine the empirical distribution of the eigenvalues and the eigenvectors of adjacency matrices of sparse regular random graphs. We find that when the degree sequence of the graph slowly increases to infinity with the number of…
We study p-Laplacians and spectral clustering for a recently proposed hypergraph model that incorporates edge-dependent vertex weights (EDVW). These weights can reflect different importance of vertices within a hyperedge, thus conferring…
The Laplacian energy of a graph is the sum of the distances of the eigenvalues of the Laplacian matrix of the graph to the graph's average degree. The maximum Laplacian energy over all graphs on $n$ nodes and $m$ edges is conjectured to be…
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…
Spectral clustering is a well-known technique which identifies $k$ clusters in an undirected graph with weight matrix $W\in\mathbb{R}^{n\times n}$ by exploiting its graph Laplacian $L(W)$, whose eigenvalues $0=\lambda_1\leq \lambda_2 \leq…
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…
Laplacian operators on finite compact metric graphs are considered under the assumption that matching conditions at graph vertices are of $\delta$ and $\delta'$ types. An infinite series of trace formulae is obtained which link together two…
Based on matrix perturbation theory, closed-form analytic expansions are studied for a Laplacian eigenvalue of an undirected, possibly weighted graph, which is close to a unique degree in that graph. An approximation is presented to provide…
We study how the spectral gap of the normalized Laplacian of a random graph changes when an edge is added to or removed from the graph. There are known examples of graphs where, perhaps counterintuitively, adding an edge can decrease the…
This work studies the classical spectral clustering algorithm which embeds the vertices of some graph $G=(V_G, E_G)$ into $\mathbb{R}^k$ using $k$ eigenvectors of some matrix of $G$, and applies $k$-means to partition $V_G$ into $k$…
An efficient spatial regularization method using superpixel segmentation and graph Laplacian regularization is proposed for sparse hyperspectral unmixing method. Since it is likely to find spectrally similar pixels in a homogeneous region,…
Spectral sparsification is a technique that is used to reduce the number of non-zero entries in a positive semidefinite matrix with little changes to its spectrum. In particular, the main application of spectral sparsification is to…
In this work, we develop a spectral theory for hypergraph limits. We prove the convergence of the spectra of adjacency and Laplacian matrices for hypergraph sequences converging in the $1$-cut metric. On the other hand, we give examples of…
We study graph-based Laplacian semi-supervised learning at low labeling rates. Laplacian learning uses harmonic extension on a graph to propagate labels. At very low label rates, Laplacian learning becomes degenerate and the solution is…
The scale and complexity of modern data sets and the limitations associated with testing large numbers of hypotheses underline the need for feature selection methods. Spectral techniques rank features according to their degree of…
Spectral graph sparsification aims to find ultra-sparse subgraphs whose Laplacian matrix can well approximate the original Laplacian eigenvalues and eigenvectors. In recent years, spectral sparsification techniques have been extensively…
It is reported that dynamical systems over digraphs have superior performance in terms of system damping and tolerance to time delays if the underlying graph Laplacian has a purely real spectrum. This paper investigates the topological…