Related papers: The Laplacian on planar graphs and graphs on surfa…
We study two different types of gluing for graphs: interface (obtained by choosing a common subgraph as the gluing component) and bridge gluing (obtained by adding a set of edges to the given subgraphs). We introduce formulae for computing…
This note discusses a relation between the multiplicity m of the second eigenvalue {\lambda}2 of a Laplacian on a graph G, tight mappings of G and a discrete analogue of Courant's nodal line theorem. For a certain class of graphs, we show…
In this paper we show a monotonicity theorem for the harmonic eigenfunction of \lambda_{1} of the normalized Laplacian over the points of articulation of a graph. We introduce the definition of Perron component for the normalized Laplacian…
We study the structure of eigenfunctions of the Laplacian on quantum graphs, with a particular focus on Morse eigenfunctions via nodal and Neumann domains. Building on Courant-type arguments, we establish upper bounds for the number of…
We present an analog to classic potential theory on weighted graphs. With nodes partitioned into exterior, boundary and interior nodes and an appropriate decomposition of the Laplacian, we define discrete analogues to the trace operators,…
We consider a nonlinear flow on simplicial complexes related to the simplicial Laplacian, and show that it is a generalization of various consensus and synchronization models commonly studied on networks. In particular, our model allows us…
The graph Laplacian is a fundamental object in the analysis of and optimization on graphs. This operator can be extended to a simplicial complex $K$ and therefore offers a way to perform ``signal processing" on $p$-(co)chains of $K$.…
We study the relationship between the symbol of the Dirichlet-to-Neumann operator associated with a connection Laplacian, and the geometry on and near the boundary. As a consequence, we show that the geometric data on the boundary, and when…
Recently, graph convolutional network (GCN) has been widely used for semi-supervised classification and deep feature representation on graph-structured data. However, existing GCN generally fails to consider the local invariance constraint…
Though commonly found in the real world, directed networks have received relatively less attention from the literature in which concerns their topological and dynamical characteristics. In this work, we develop a magnetic Laplacian-based…
Graphs with diverse structural characteristics play a central role in modelling and optimization tasks. The ability to generate different types of graphs that exhibit shared properties is likewise essential for algorithm selection and…
We consider Laplace operators on periodic discrete graphs perturbed by guides, i.e., graphs which are periodic in some directions and finite in other ones. The spectrum of the Laplacian on the unperturbed graph is a union of a finite number…
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…
We develop theory and methods that use the graph Laplacian to analyze the geometry of the underlying manifold of datasets. Our theory provides theoretical guarantees and explicit bounds on the functional forms of the graph Laplacian when it…
We establish several new relations between the discrete transition operator, the continuous Laplacian and the averaging operator associated with combinatorial and metric graphs. It is shown that these operators can be expressed through each…
Recently, much of the existing work in manifold learning has been done under the assumption that the data is sampled from a manifold without boundaries and singularities or that the functions of interest are evaluated away from such points.…
Higher-order networks have so far been considered primarily in the context of studying the structure of complex systems, i.e., the higher-order or multi-way relations connecting the constituent entities. More recently, a number of studies…
For a given infinite connected graph $G=(V,E)$ and an arbitrary but fixed conductance function $c$, we study an associated graph Laplacian $\Delta_{c}$; it is a generalized difference operator where the differences are measured across the…
The Laplacian eigenvalues of a network play an important role in the analysis of many structural and dynamical network problems. In this paper, we study the relationship between the eigenvalue spectrum of the normalized Laplacian matrix and…
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,…