arXiv research

Maximal knotless graphs have at least 74% of their vertices' edges.

Fewer obstructions for small graphs in knotless embedding.

New constructions from non-separating planar graphs improve understanding of graph linkability and knotability.

We describe an algorithm that recognizes some (perhaps all) intrinsically knotted (IK) graphs, and can help find knotless embeddings for graphs that are not IK. The algorithm, implemented as a Mathematica program, has already been used by Goldberg, Mattman, and Naimi [6] to greatly expand the list of known minor minima…

3028 obstructions found for embedding without knots.

We show that the minimum number of sticks required to construct a non-paneled knotless embedding of $K_4$ is 9 and of $K_5$ is 12 or 13. We use our results about $K_4$ to show that the probability that a random linear embedding of $K_{3,3}$ in a cube is in the form of a Möbius ladder is $0.97380\pm 0.00003$, and offer …

New bounds on maximal linkless graphs with improved edge-to-vertex ratios.

A maximally linkless graph is a graph that can be embedded in $\mathbb{R}^3$ without any links, but cannot be embedded in such a way if any other edge is added to the graph. Recently, a family of maximally linkless graphs was found with $m=3n-3$ edges. We improve upon this by demonstrating a new family of maximally lin…

The paper shows conflict graphs of Petersen family graphs are mostly unbalanced.

The study extends Tutte's conflict graph concept to nonplanar graphs.

Maximal diameter theorem for graphs with positive Ricci curvature.

Study finds maximal linklessly embeddable graphs up to 11 vertices and their complements.

Constructs graphs with singularities in a special space.

Maximal surfaces in Lorentz-Minkowski space have conjugate graphs.

We extend Osserman's lemma on the generalized Gauss map of two-dimensional minimal graphs of higher codimension, construct a Jenkins-Serrin type special Lagrangian Scherk graph explicitly, and generalize Calabi's correspondence between minimal graphs and maximal graphs.

We show that the number of entire maximal graphs with finitely many singular points that are conformally equivalent is a universal constant that depends only on the number of singularities, namely 2^$ for graphs with n+1 singularities. We also give an explicit description of the family of entire maximal graphs with a f…

In the lorentzian product $\Bbb G^n\times\Bbb R_1,$ we give a comparison between the $f$-volume of an entire $f$-maximal graph and the $f$-volume of the hyperbolic $H_r^+$ under the assumption that the gradient of the function defining the graph is bounded away from 1. As a consequence, we obtain a Bernstein type theor…

Algorithm calculates genus of embedded graphs on surfaces.

New simplicial complex for infinite-type surfaces shows graph properties.

In this paper, we unify the Markov theory of a variety of different types of graphs used in graphical Markov models by introducing the class of loopless mixed graphs, and show that all independence models induced by $m$-separation on such graphs are compositional graphoids. We focus in particular on the subclass of rib…

New method for fair influence maximization in social networks.

New method constructs graphs from data efficiently, suitable for large datasets.

GraphCL learns node representations by maximizing similarity between perturbed node features.

Proposes a novel graph self-training method with EM regularization for semi-supervised node classification.

If the Lorentzian norm on a maximal surface in the 3-dimensional Lorentz-Minkowski space $R_1^3$ is positive and proper, then the surface is relative parabolic. As a consequence, entire maximal graphs with a closed set of isolated singularities are relative parabolic. Furthermore, maximal and minimal graphs over closed…

This work improves GNN training efficiency by maximizing ego-graph information.

New theorem connects minimal and maximal surfaces, affecting graphness.

A novel method integrates feature and topology views for unsupervised graph representation learning.

We construct a partial order relation which acts on the set of 3-cliques of a maximal planar graph G and defines a unique hierarchy. We demonstrate that G is the union of a set of special subgraphs, named `bubbles', that are themselves maximal planar graphs. The graph G is retrieved by connecting these bubbles in a tre…

A new algorithm learns MAGs from data more efficiently using entropy.

We show that a Born-Infeld soliton can be realised either as a spacelike minimal graph or timelike minimal graph over a timelike plane or a combination of both away from singular points. We also obtain some exact solutions of the Born-Infeld equation from already known solutions to the maximal surface equation. Further…

Iterative Proportional Fitting (IPF), combined with EM, is commonly used as an algorithm for likelihood maximization in undirected graphical models. In this paper, we present two iterative algorithms that generalize upon IPF. The first one is for likelihood maximization in discrete chain factor graphs, which we define …

The main goal of this survey is to illustrate geometric applications of the Poincaré Lemma to constant mean curvature equations. In 1970, Calabi introduced the duality between minimal graphs in three dimensional Euclidean space and maximal graphs in three dimensional Lorentz space. We construct two extensions of Calabi…

A variety of graph neural networks (GNNs) frameworks for representation learning on graphs have been recently developed. These frameworks rely on aggregation and iteration scheme to learn the representation of nodes. However, information between nodes is inevitably lost in the scheme during learning. In order to reduce…

In this paper we establish some parabolicity criteria for maximal surfaces immersed into a Lorentzian product space of the form $M^2\times\mathbb{R}_1$, where $M^2$ is a connected Riemannian surface with non-negative Gaussian curvature and $M^2\times\mathbb{R}_1$ is endowed with the Lorentzian product metric $<,>=<,>_M…

New GCNs solve graph embedding problems efficiently and interpretably.

Spatio-temporal graphs such as traffic networks or gene regulatory systems present challenges for the existing deep learning methods due to the complexity of structural changes over time. To address these issues, we introduce Spatio-Temporal Deep Graph Infomax (STDGI)---a fully unsupervised node representation learning…

MissNODAG learns cyclic causal graphs from incomplete data.

Convolution operations designed for graph-structured data usually utilize the graph Laplacian, which can be seen as message passing between the adjacent neighbors through a generic random walk. In this paper, we propose PAN, a new graph convolution framework that involves every path linking the message sender and recei…

We consider the Dirichlet boundary value problem for graphical maximal submanifolds inside Lorentzian type ambient spaces, and obtain general existence and uniqueness results which apply to any codimension.

Graph products inherit Morse local-to-global property from their components.

Maximizes mixing efficiency in surface braids.

This paper studies learning the representations of whole graphs in both unsupervised and semi-supervised scenarios. Graph-level representations are critical in a variety of real-world applications such as predicting the properties of molecules and community analysis in social networks. Traditional graph kernel based me…

A new method improves graph node embeddings by considering both nearby and distant node similarities.

The richness in the content of various information networks such as social networks and communication networks provides the unprecedented potential for learning high-quality expressive representations without external supervision. This paper investigates how to preserve and extract the abundant information from graph-s…

Unsupervised method learns hierarchical graph representations without labels.

Graph clustering improved using Boltzmann machine heuristics.

Graph Neural Networks (GNNs) achieve an impressive performance on structured graphs by recursively updating the representation vector of each node based on its neighbors, during which parameterized transformation matrices should be learned for the node feature updating. However, existing propagation schemes are far fro…