Simple graphs with 12 nodes and 6 neighbors always have a 6-node subgraph.
problem Finding a specific subgraph in simple graphs.
method Proving every graph of order 12 with minimum degree 6 contains a K_6 minor.
result Simple graphs of order 12 and minimum degree 6 contain K_6 minors.
For graphs with 13 or more vertices, either the graph or its complement is intrinsically linked.
problem Identifying graphs with 13 or more vertices that are intrinsically linked or their complements.
method Analyzing properties of graphs and their complements to determine if either is intrinsically linked.
result For graphs with 13 or more vertices, either the graph or its complement is intrinsically linked.
Simple graph representation outperforms complex methods in graph classification.
problem Graph classification and representation learning on graphs.
method Developed a simple yet meaningful graph representation and tested its effectiveness.
result Simple graph representation achieves similar performance to state-of-the-art methods for non-attributed graph classification.
Simple proof shows graph neural networks are versatile.
problem Proving the universality of graph neural networks.
method Introduced a Graph Homomorphism Model to prove universality.
result Simple proofs of graph neural network universality.
New graph shows edge deletion/contraction doesn't always result in intrinsically linked graphs.
problem Edge operations in intrinsically knotted graphs don't always produce intrinsically linked graphs.
method Presented a new intrinsically knotted graph.
result Edge operations in intrinsically knotted graphs don't always result in intrinsically linked graphs.
Simple algorithm outperforms complex methods in graph classification.
problem Efficient graph classification methods with comparable performance to state-of-the-art algorithms.
method Spectral decomposition of graph Laplacian.
result Simple algorithm achieves competitive results.
This paper classifies chiral graphs up to size 12.
problem Understanding the chirality of simple graphs to predict molecular behavior.
method Classifying minor minimal intrinsically chiral graphs among simple graphs of size up to 12.
result Complete set of minor minimal graphs for intrinsic properties of chiral molecules.
Simple linear model outperforms GCN in graph AE tasks.
problem Challenging tasks like link prediction and node clustering.
method Replaced GCN with a simple linear model on adjacency matrix.
result Simple model consistently reaches competitive performances.
ELD compares graphs by their embedded Laplacian eigenvectors, resolving ambiguities.
problem Comparing graphs of different sizes and structures.
method ELD uses symmetrization and perturbation techniques to compare graph embeddings.
result ELD resolves ambiguities in graph comparisons, making it a natural pseudo-metric.
Automorphisms of fine 1-curve graph linked to surface homeomorphisms.
problem Understanding automorphisms of fine 1-curve graphs.
method Isomorphic mapping to surface homeomorphisms.
result Automorphism group is isomorphic to homeomorphism group of a surface.
The paper characterizes graph manifolds using fold maps and embeddability of polyhedra.
problem Understanding the global topologies of graph manifolds.
method Using fold maps into the plane and embeddability of polyhedra in 3-manifolds.
result Characterizes graph manifolds via fold maps and polyhedra embeddability.
The maximum number of maximum cliques in a graph is determined for graphs with at least 15 vertices.
problem Determining the maximum number of maximum cliques in a graph with n vertices.
method Defining prime and composite graphs, analyzing edge bounds, and using combinatorial arguments.
result For graphs with at least 15 vertices, the graph with the maximum number of maximum cliques is composite.
GraphACL learns graph representations without augmentation or homophily assumptions.
problem Learning graph representations on heterophilic graphs (nodes with different labels and features).
method Asymmetric Contrastive Learning for Graphs (GraphACL) considers an asymmetric view of neighboring nodes.
result GraphACL significantly outperforms state-of-the-art methods on both homophilic and heterophilic graphs.
Random subsurfaces of hyperbolic surfaces equidistribute to ribbon graphs.
problem Distribution of shapes of complementary subsurfaces in moduli space.
method Study of shapes of complementary subsurfaces in moduli space as boundary lengths go to infinity.
result Random subsurfaces look like random ribbon graphs.
A regularized optimization problem over a large unstructured graph is studied, where the regularization term is tied to the graph geometry. Typical regularization examples include the total variation and the Laplacian regularizations over the graph. When applying the proximal gradient algorithm to solve this problem, t…
The paper examines deformations of simple dotted graphs made of circles.
problem Investigating reducibility of admissible dotted graphs.
method Analyzes deformations of admissible dotted graphs consisting of standard circles.
result Identifies specific conditions under which certain dotted graphs can be reduced.
Knowledge graphs contain knowledge about the world and provide a structured representation of this knowledge. Current knowledge graphs contain only a small subset of what is true in the world. Link prediction approaches aim at predicting new links for a knowledge graph given the existing links among the entities. Tenso…
Rotation systems can't always be drawn in surfaces.
problem Rotation systems and simple drawings in surfaces.
method Extended the plane result to all fixed surfaces.
result Existence of rotation systems not arising from simple drawings in any fixed surface.
The paper calculates graph Ricci curvature and finds properties of specific graph types.
problem Understanding Ricci curvature on irregular graphs.
method Developed a formula for graph Ricci curvature based on optimal bijections.
result Derived structural and theorem results for specific graph types.
Connected graph for twice-punctured torus curves.
problem Structure of tri-pants graph on twice-punctured torus.
method Examined relationship with Farey complex to prove connectivity and infinite diameter.
result Tri-pants graph is connected and has infinite diameter.
New method to classify simple Smale flows on S3.
problem Classifying simple Smale flows on S3. method Embedded template and Kauffman's invariant of spatial graphs.
result Isotopic classification of simple Smale flows on S3. Study Morse functions on projective plane using Reeb graphs.
problem Investigate topological structure of Morse functions on projective plane.
method Use Reeb graphs to describe and prove properties of simple Morse functions on RP2. result Prove that Reeb graphs are a complete topological invariant for simple Morse functions on RP2. Machine learning techniques have recently been adopted in various applications in medicine, biology, chemistry, and material engineering. An important task is to predict the properties of molecules, which serves as the main subroutine in many downstream applications such as virtual screening and drug design. Despite th…
New algorithms learn simple staged trees from data, improving model fit.
problem Complex conditional independences in categorical data vectors.
method Structural learning algorithms for simple staged trees, coalescing the underlying tree.
result Data-learned simple staged trees often outperform Bayesian networks in model fit.
New non-homophilous graph datasets and methods for scalable learning.
problem Evaluation of graph learning methods on non-homophilous graphs.
method Introducing LINKX, a simple yet strong method for scalable non-homophilous graph learning.
result LINKX achieves state-of-the-art performance on non-homophilous graphs.
Graph convolutional networks fail to use eigenvectors beyond the first, unlike spectral embedding.
problem Understanding when graph convolutional networks fail compared to spectral embedding.
method Presented a simple generative model to illustrate failure.
result Graph convolutional networks fail to use eigenvectors beyond the first in certain graphs.
Simple Euclidean models outperform hyperbolic graph learning models.
problem The effectiveness of hyperbolic graph learning models is questioned.
method Careful analysis of hyperbolic graph representation learning, identifying and addressing issues with baselines, modeling assumptions, and metric usage.
result Simple Euclidean models often outperform hyperbolic graph learning models, even on hyperbolic datasets.
Graph Neural Networks solve topology problems in simple 3D models.
problem Deciding homeomorphism of 3-manifolds described by plumbing graphs.
method Supervised and reinforcement learning with Graph Neural Networks.
result High accuracy in determining homeomorphic 3-manifolds.
A zigzag in a plane graph is a circuit of edges, such that any two, but no three, consecutive edges belong to the same face. A railroad in a plane graph is a circuit of hexagonal faces, such that any hexagon is adjacent to its neighbors on opposite edges. A graph without a railroad is called tight. We consider the zigz…
In this paper, we study classes of graphs with three types of edges that capture the modified independence structure of a directed acyclic graph (DAG) after marginalisation over unobserved variables and conditioning on selection variables using the m-separation criterion. These include MC, summary, and ancestral grap…
Determinants of theta curves and symmetric graphs are studied.
problem Understanding the determinants of theta curves and symmetric graphs.
method Combinatorial approach using Kirchhoff's Matrix Tree Theorem and spanning tree enumeration.
result The determinant of a simple theta curve is the product of the determinants of its constituent knots.
Simple algorithm samples graph nodes efficiently.
problem Sampling representative nodes from large graphs efficiently.
method Minimum inner product greedy selection rule, column-selective sampling.
result Achieves sampling proportional to cluster size, error decays with inter-cluster connectivity.
The paper studies actions on Bass-Serre trees and identifies new C∗-simple groups.
problem Investigating actions of fundamental groups on Bass-Serre trees and their C∗-algebraic properties. method Analyzing boundary actions of fundamental groups of graphs of groups on their Bass-Serre trees.
result Identification of new families of C∗-simple groups, including tubular groups and certain graphs of groups. A new graph generation model uses Mallat's scattering transform.
problem Unclear mathematical properties and difficulty in training good generative models for graphs.
method Proposes a graph generation model using a Gaussianized graph scattering transform.
result Demonstrates state-of-the-art performance in link prediction and graph/signal generation.
Graph classification models are sensitive to initialisation and structure, but simple models perform well.
problem Graph classification models' performance is sensitive to initialisation and structure.
method Examined recent graph coarsening architectures and their performance sensitivity.
result Simple models like MLP, single-layer GCN, and fixed-weight GCN achieve competitive performance.
Classifies graph configuration spaces homeomorphic to manifolds.
problem Classifying graph configuration spaces homeomorphic to manifolds.
method Developed techniques to translate topological properties into graph theoretic ones.
result Extended Abrams' work to classify certain graph configuration spaces.
GNNs are powerful but limited in their ability to distinguish certain graph structures.
problem Limited understanding of GNNs' representational properties and limitations.
method Theoretical framework and analysis of GNN expressive power, development of a provably most expressive GNN architecture.
result GNNs cannot learn to distinguish certain simple graph structures, but a new architecture can.
SASE improves attributed graph clustering for large graphs with linear time and space complexity.
problem Challenges in clustering large attributed graphs due to high computational and memory costs.
method SASE combines node features smoothing, scalable spectral clustering, and adaptive order selection.
result SASE achieves a 6.9% improvement in ACC and a 5.87x speedup on the ArXiv dataset.
We consider a method popular in the literature of associating a two-step nilpotent Lie algebra with a finite simple graph. We prove that the two-step nilpotent Lie algebras associated with two graphs are Lie isomorphic if and only if the graphs from which they arise are isomorphic.
New graph neural networks can distinguish graphs better than previous models.
problem Graph isomorphism tests limit the expressive power of GNNs.
method Developed k-order invariant and equivariant graph neural networks, and a reduced 2-order network.
result A reduced 2-order network with a single quadratic operation has 3-WL expressiveness, surpassing message passing models.
Explains partial duality for ribbon graphs in simple terms.
problem Understanding partial duality in ribbon graphs.
method Simplified explanation of existing research on hypermaps.
result Simplified exposition of complex graph theory concepts.
RL-VAE uses RL to decode molecular graphs from latent embeddings.
problem Efficiently decoding molecular graphs from latent embeddings.
method Repurposed simple graph generator for efficient decoding.
result Decoding molecular graphs from latent embeddings is possible with a simple graph generator.
Refines Ozsváth-Szabó d-invariants for knot concurrence.
problem Computing Ozsváth-Szabó d-invariants for specific knot types.
method Refines and applies Karakurt and Şavk's formula for surgeries on almost simple linear graphs.
result Inequalities for d-invariants are equalities or strict in specific families.
In the present paper we construct a one-to-one correspondence between the set of graph-knots and the set of homotopy classes of looped graphs. Moreover, the graph-knot and the homotopy class constructed from a given knot are related with this correspondence. This correspondence is given by a simple formula.
Proves a theorem connecting graph theory spheres, reformulating Morse conditions.
problem Defines and connects spheres in graph theory.
method Proves a theorem bridging two graph theory sphere definitions.
result Reformulates Morse conditions using center manifolds and level surface graphs.
We present a graph manifold analog of the Jankins-Neumann classification of Seifert fibered spaces over S2 admitting taut foliations, providing a finite recursive formula to compute the L-space Dehn-filling interval for any graph manifold with torus boundary. As an application of a generalization of this result to F…
Graph neural networks perform better with more features and training data.
problem Comparing graph neural networks to simple models in semi-supervised node classification.
method Empirical evaluation of graph neural network architectures in various settings.
result More complex graph networks outperform simple models in settings with fewer features and more training data.
edGNN improves graph embeddings for directed labeled graphs.
problem Improving node and graph embeddings for directed labeled graphs.
method edGNN is a GNN designed for directed labeled graphs, leveraging both topology and labels.
result edGNN is as powerful as the Weisfeiler-Lehman algorithm for graph isomorphism.