A maximally linkless graph is a graph that can be embedded in 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 edges. We improve upon this by demonstrating a new family of maximally lin…
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48 results for “linkless graphs”
Notes a flaw in a proof about embedding graphs.
problem A flaw in proving Sachs' conjecture about graph embeddings.
method Analyzing Stanfield's proof for gaps.
result Identifies a significant error in the proof.
Classifies intrinsically linked tournaments by their score sequences.
problem Classifying intrinsically linked tournaments using their score sequences.
method Examining the score sequences of tournaments and identifying linkless sequences.
result The vast majority of score sequences for 8-vertex tournaments are linkless.
New bounds on maximal linkless graphs with improved edge-to-vertex ratios.
problem Finding maximal linklessly embeddable graphs with improved edge-to-vertex ratios.
method Constructing families of graphs and proving necessary and sufficient conditions for clique sums.
result Improved edge-to-vertex ratios for maximal linklessly embeddable graphs.
Graphs embeddable on torus and linklessly in 3D can be embedded linklessly in standard torus.
problem Embedding linklessly in a standard torus for graphs embeddable on torus and in 3D.
method Analyzing graphs of order 9 and below, showing linkless embedding in standard torus.
result For graphs of order 9 and below, linkless embedding in standard torus is possible.
New graphs found that can be drawn without crossing links.
problem Finding graphs that can be drawn without links crossing.
method Provided specific examples for each .
result Infinite family of graphs linklessly embeddable and Tutte-4-connected.
A simpler proof for apex graphs in McCarty and Thomas' conjecture.
problem Proving a conjecture about apex graphs and their linklessly embeddable properties.
method Shorter and simpler proof for the apex case.
result A shorter and simpler proof for the apex case of the conjecture.
We announce results about flat (linkless) embeddings of graphs in 3-space. A piecewise-linear embedding of a graph in 3-space is called {\it flat} if every circuit of the graph bounds a disk disjoint from the rest of the graph. We have shown: (i) An embedding is flat if and only if the fundamental group of the compleme…
Constructions stemming from non-separating planar graphs and their Colin de Verdière invariantmath.CO
New constructions from non-separating planar graphs improve understanding of graph linkability and knotability.
problem Understanding linkability and knotability of graph complements.
method Using maximal non-separating planar graphs to construct examples of maximal linkless and knotless graphs, and analyzing their Colin de Verdière invariant.
result The Colin de Verdière invariant of the complement of a maximal non-separating planar graph satisfies μ(cG) ≤ n-4, and equality holds.
Study finds maximal linklessly embeddable graphs up to 11 vertices and their complements.
problem Characterizing linklessly embeddable graphs and their complements.
method Comprehensive search and verification of graphs up to 11 vertices.
result For graphs of order 11, either the graph or its complement is intrinsically linked.
We consider intrinsic linking and knotting in the context of directed graphs. We construct an example of a directed graph that contains a consistently oriented knotted cycle in every embedding. We also construct examples of intrinsically 3-linked and 4-linked directed graphs. We introduce two operations, consistent edg…
Study embeddability of 2-complexes in 4-space, proving Heawood family's excluded minors.
problem Whether a 2-dimensional CW complex embeds in .
method Operations preserving embeddability, constructions of non-preserving transformations, study of 4-flat graphs.
result Prove 78 graphs of Heawood family are excluded minors for 4-flat graphs.
The dominant graph neural networks (GNNs) over-rely on the graph links, several serious performance problems with which have been witnessed already, e.g., suspended animation problem and over-smoothing problem. What's more, the inherently inter-connected nature precludes parallelization within the graph, which becomes …
We review a cochain-free treatment of the classical van Kampen obstruction θto embeddability of an n-polyhedron into R^{2n} and consider several analogues and generalizations of θ, including an extraordinary lift of θwhich in the manifold case has been studied by J.-P. Dax. The following results are obtained. - The mod…
Line graph transformation aids graph isomorphism tests by excluding challenging graph properties.
problem Limited theoretical understanding of line graph transformation's impact on GNN models.
method Examined CFI and strongly regular graphs, showing line graph transformation helps WL tests distinguish these graphs.
result Line graph transformation aids WL tests in distinguishing challenging graph properties.
Proposes MGMN for end-to-end graph similarity learning.
problem Lack of cross-level interactions in graph similarity learning.
method Multi-level graph matching network (MGMN) combining node-graph matching and siamese graph neural networks.
result MGMN outperforms state-of-the-art models on graph-graph classification and regression tasks.
Graphons of Line Graphsstat.ML
The paper explores graphons of line graphs from sparse finite graphs.
problem Estimating graph limits from sparse finite graphs.
method Mapping finite graphs to their line graphs and analyzing graphs with the square-degree property.
result Graphons of line graphs can distinguish between sparse graphs like star graphs and superlinear preferential attachment graphs.
MxPool learns graph features from diverse graphs using a hierarchical structure.
problem Learning graph features from diverse graphs with varying properties and sizes.
method MxPool uses a multiplex structure with multiple graph convolution/pooling networks in a hierarchical learning structure.
result MxPool outperforms state-of-the-art methods on graph classification benchmarks.
Study the geometry of graph product extension graphs.
problem Properties of graph products.
method Introduce and study the extension graph of graph products of groups.
result Extension graph is isomorphic to crossing graph of a quasi-median graph and exhibits asymptotic dimension similar to quasi-trees.
Graph neural network learns graph distances effectively.
problem Maintaining graph distance metric properties.
method GRAPH-BERT based semi-supervised distance metric learning.
result GB-DISTANCE outperforms existing methods.
Quasi-transitive graphs quasi-isometric to planar graphs can be upgraded to Cayley graphs.
problem Quasi-transitive graphs quasi-isometric to planar graphs need to be upgraded to Cayley graphs.
method Upgrading a planar graph to a Cayley graph.
result Quasi-transitive graphs quasi-isometric to planar graphs can be upgraded to Cayley graphs.
Customized-GNN generates model-specific for each graph.
problem Graphs in the same dataset have distinct structures.
method Proposes Customized-GNN framework to generate model-specific for each graph.
result Demonstrates effectiveness on various graph classification benchmarks.
Graph embedding leaks sensitive graph properties and subgraphs.
problem Privacy risks in graph embedding sharing.
method Three inference attacks and a defense mechanism.
result High accuracy in inferring graph properties and subgraphs.
Characterizes graphs with leveled embeddings and introduces new graph invariants.
problem Understanding the properties of leveled embeddings in spatial graphs.
method Characterization of graphs with leveled embeddings, introduction of new invariants.
result Characterization of graphs with low level number and determination of specific invariants for complete graphs and complete bipartite graphs.
The paper shows conflict graphs of Petersen family graphs are mostly unbalanced.
problem Understanding the balance of conflict graphs in Petersen family graphs.
method Analyzing maximally planar subgraphs and their conflict graphs.
result All but three strong conflict graphs from Petersen Family Graphs are unbalanced.
Two new methods improve graph embedding without needing a complete graph structure.
problem Graph autoencoders' performance depends on the adjacency matrix quality.
method BAGE and VBAGE: unsupervised graph embedding via adaptive graph learning.
result The methods expand GAEs' applicability to datasets without graph structure.
We define a pseudo-inverse for line graphs using linear integer programming.
problem Not all graphs have a corresponding root graph, making the line graph operation non-invertible.
method Propose a linear integer program to edit the smallest number of edges in the line graph to recover a root graph.
result The pseudo-inverse operation is well-behaved and works in practice as shown by empirical experiments.
Develops method to create non-Abelian Ricci-flat graphs via bundles.
problem Creating non-Abelian Ricci-flat graphs.
method Develops systematic way via graph bundles with constraints.
result Non-trivial graph bundles are not isomorphic to product of base and fiber.
New method uses graph generative models for graph classification.
problem Graph classification for non-relational i.i.d. data.
method Derive classification formulas from GGM, train generative graph auto-encoder model.
result New conditional ELBO for training graph auto-encoder model.
MathNet uses wavelets for graph representation and learning.
problem Graph Neural Networks (GNNs) for graph classification and regression.
method Multiresolution Haar-like wavelets, graph convolution, and pooling.
result MathNet achieves notable accuracy gains on graph classification and regression tasks.
Unified framework for graph coarsening using node features and graph matrices.
problem Dimensionality reduction of large graphs while preserving node features.
method Optimization-based framework that unifies graph learning and dimensionality reduction.
result The learned coarsened graph is ε-similar to the original graph, where ε is a small positive number.
A fast graph embedding method for large graphs.
problem Efficiently embedding large graphs for various applications.
method One-hot graph encoder embedding with linear complexity.
result Graph encoder embedding is approximately normally distributed and converges to its mean.
Quadratic bounds found for graph dimensions.
problem Understanding dimensions of arc and disk graphs.
method Quadratic upper bounds calculation.
result Asymptotic dimensions of arc and disk graphs have been bounded.
Study classifies Halin graphs with positive curvature.
problem Classifying Halin graphs with specific curvature.
method Analyzing generalized Halin graphs formed by connecting tree leaves.
result Identified all generalized Halin graphs with positive Lin-Lu-Yau curvature.
PSimGNN partitions graphs into subgraphs for efficient graph similarity computation.
problem Efficiently compute graph similarity scores for large graphs.
method Graph partitioning followed by subgraph-level and node-level comparisons using a graph neural network.
result PSimGNN outperforms state-of-the-art methods in graph similarity computation tasks.
We present graph wavelet neural network (GWNN), a novel graph convolutional neural network (CNN), leveraging graph wavelet transform to address the shortcomings of previous spectral graph CNN methods that depend on graph Fourier transform. Different from graph Fourier transform, graph wavelet transform can be obtained …
This paper develops graph theory for racks and quasigroups.
problem Characterizing and realizing right quasigroups and related structures.
method Study of graph markings, Schreier graphs, and Cayley graphs.
result All right quasigroups are realizable by specific types of graphs.
Graph Convolutional Neural Networks (Graph CNNs) are generalizations of classical CNNs to handle graph data such as molecular data, point could and social networks. Current filters in graph CNNs are built for fixed and shared graph structure. However, for most real data, the graph structures varies in both size and con…
We introduce a novel approach to graph-level representation learning, which is to embed an entire graph into a vector space where the embeddings of two graphs preserve their graph-graph proximity. Our approach, UGRAPHEMB, is a general framework that provides a novel means to performing graph-level embedding in a comple…
Graph Cascades rewire graphs to improve structure-aware learning.
problem Improving graph neural networks and transformers for structure-aware learning.
method Graph Cascades uses contagion-based diffusion processes to construct an auxiliary graph with reinforced edges.
result Graph Cascades improves node-classification benchmarks across various graph types.
CTGCN learns dynamic graph embeddings preserving both local and global graph structure.
problem Learning node representations for evolving graphs while preserving both local and global graph structure.
method CTGCN uses k-core based temporal graph convolutional network to learn dynamic graph embeddings.
result CTGCN outperforms existing methods in link prediction and structural role classification.
Graphs can be fooled by small edge changes, but this work protects them.
problem Adversaries can manipulate graph data to mislead graph classification models.
method We introduce a smoothed graph classification model with a robustness guarantee.
result The smoothed model maintains consistent predictions under small adversarial perturbations.
The study bounds the effective diameter of graphs with positive Ollivier curvature.
problem Bounding the effective diameter of graphs with positive Ollivier curvature.
method Introducing reflective graphs and proving discrete Bonnet Myers theorem.
result The effective diameter bound is attained only for specific graphs.
SpeqNets improve graph neural networks by scaling and adapting to graph sparsity.
problem Graph neural networks struggle with permutation-equivariant functions and scalability to large graphs.
method Introducing sparsity-aware, permutation-equivariant graph networks with heuristics for graph isomorphism.
result Significantly improved predictive performance and reduced computation times compared to existing methods.
Paper converts graph learning to lifelong learning.
problem Learning graphs in a streaming fashion.
method Feature graph topology, converting node classification to graph classification.
result FGN achieves superior performance in lifelong human action recognition and feature matching.
Graph embedding is an effective method to represent graph data in a low dimensional space for graph analytics. Most existing embedding algorithms typically focus on preserving the topological structure or minimizing the reconstruction errors of graph data, but they have mostly ignored the data distribution of the laten…
Bochner's Identity on Graphsmath.DG
Proves Bochner's identity on graphs using a new auxiliary graph.
problem Extending Bochner's identity to graph theory.
method Introduces a complete tangent graph to prove the identity.
result Validates Bochner's identity on graphs.
AgraSSt assesses graph generators using Stein operators and kernel discrepancies.
problem Assessing the quality of graph generators that are implicit or not in explicit form.
method AgraSSt uses Stein operators and kernel discrepancies to assess graph generators, providing interpretable criticisms.
result Theoretical guarantees and empirical validation for various graph models.