Related papers: Extremal graph realizations and graph Laplacian ei…

In recent years, spectral graph neural networks, characterized by polynomial filters, have garnered increasing attention and have achieved remarkable performance in tasks such as node classification. These models typically assume that…

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 consider the problem of finding universal bounds of "isoperimetric" or "isodiametric" type on the spectral gap of the Laplacian on a metric graph with natural boundary conditions at the vertices, in terms of various analytical and…

Let $G$ be a connected simple graph with $n$ vertices. The distance Laplacian matrix $D^{L}(G)$ is defined as $D^L(G)=Diag(Tr)-D(G)$, where $Diag(Tr)$ is the diagonal matrix of vertex transmissions and $D(G)$ is the distance matrix of $G$.…

We are concerned with spectral problems of the Goldberg-Coxeter construction for $3$- and $4$-valent finite graphs. The Goldberg-Coxeter constructions $\mathrm{GC}_{k,l}(X)$ of a finite $3$- or $4$-valent graph $X$ are considered as…

We apply eigenvalue interlacing techniques for obtaining lower and upper bounds for the sums of Laplacian eigenvalues of graphs, and characterize equality. This leads to generalizations of, and variations on theorems by Grone, and Grone and…

Let $G$ be an irregular graph on $n$ vertices with maximum degree $\Delta$ and diameter $D$. We show that \Delta-\lambda_1>\frac{1}{nD} where $\lambda_1$ is the largest eigenvalue of the adjacency matrix of $G$. We also study the effect of…

We investigate properties of the sequences of extremal values that could be achieved by the eigenvalues of the Laplacian on Euclidean domains of unit volume, under Dirichlet and Neumann boundary conditions, respectively. In a second part,…

In this note, we give an infinite family of optimal graphs called $G^+(d,c)$. They are optimal in the sense that they have the maximum possible number of vertices for given a diameter $d$ and the so-called `outer multiset dimension' $c$. We…

This paper presents the applications of Eigenvalues and Eigenvectors (as part of spectral decomposition) to analyze the bipartivity index of graphs as well as to predict the set of vertices that will constitute the two partitions of graphs…

The isoperimetric problem of maximizing all eigenvalues of the Laplacian on a metric tree graph within the class of trees of a given average edge length is studied. It turns out that, up to rescaling, the unique maximizer of the $k$-th…

Our primary motivation is existence and uniqueness for the obstacle problem on graphs. That is, we look for unique solutions to the problem $Lu = \chi_{\{u>0\}}$, where $L$ is the Laplacian matrix associated to a graph, and $u$ is a…

In this paper we study Lipschitz regularity of elliptic PDEs on geometric graphs, constructed from random data points. The data points are sampled from a distribution supported on a smooth manifold. The family of equations that we study…

The finite element method(FEM) is applied to bound leading eigenvalues of Laplace operator over polygonal domain. Compared with classical numerical methods, most of which can only give concrete eigenvalue bounds over special domain of…

Manifold learning methods play a prominent role in nonlinear dimensionality reduction and other tasks involving high-dimensional data sets with low intrinsic dimensionality. Many of these methods are graph-based: they associate a vertex…

Eigenmaps are important in analysis, geometry, and machine learning, especially in nonlinear dimension reduction. Approximation of the eigenmaps of a Laplace operator depends crucially on the scaling parameter $\epsilon$. If $\epsilon$ is…

The graph entropy describes the structural information of graph. Motivated by the definition of graph entropy in general graphs, the graph entropy of hypergraphs based on Laplacian degree are defined. Some results on graph entropy of simple…

In this paper, we show that the largest Laplacian H-eigenvalue of a $k$-uniform nontrivial hypergraph is strictly larger than the maximum degree when $k$ is even. A tight lower bound for this eigenvalue is given. For a connected…

The principal ratio of a graph is the ratio of the greatest and least entry of its principal eigenvector. Since the principal ratio compares the extreme values of the principal eigenvector it is sensitive to outliers. This can be…

How to draw the vertices of a complete multipartite graph $G$ on different points of a bounded $d$-dimensional integer grid, such that the sum of squared distances between vertices of $G$ is (i) minimized or (ii) maximized? For both…