We study the Thurston-Bennequin number of complete and complete bipartite Legendrian graphs. We define a new invariant called the total Thurston-Bennequin number of the graph. We show that this invariant is determined by the Thurston-Bennequin numbers of 3-cycles for complete graphs and by the Thurston-Bennequin number…
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48 results for “complete graph $K_4$”
This study examines how removing edges from complete graphs affects Ollivier Ricci curvature.
problem Conditions under which Ollivier Ricci curvature changes sign after edge removal.
method Defined and analyzed graphs obtained by removing matching, vertex incident, and cycle edges from complete graphs.
result Ollivier Ricci curvature remains positive or zero for graphs formed by removing edges from complete graphs.
Self-complementary graphs have complete minors.
problem Finding complete minors in self-complementary graphs.
method Analyzing topological properties of self-complementary graphs.
result Self-complementary graphs contain minors.
We say that a graph is intrinsically knotted or completely 3-linked if every embedding of the graph into the 3-sphere contains a nontrivial knot or a 3-component link any of whose 2-component sublink is nonsplittable. We show that a graph obtained from the complete graph on seven vertices by a finite sequence of $\tria…
We consider the flow on complete non-compact graphs. We prove that a complete graph evolves by the curvature up to some time depending on the radius of a sphere enclosed by the initial graph.
The paper generalizes linking number properties for complete graphs.
problem Understanding linking numbers in spatial complete graphs.
method Analyzing the sum of square linking numbers and triangle-triangle links.
result Explicit formula for the sum of square linking numbers in large complete graphs.
We study the evolution of convex complete non-compact graphs by positive powers of Gauss curvature. We show that if the initial complete graph has a local uniform convexity, then the graph evolves by any positive power of Gauss curvature for all time. In particular, the initial graph is not necessarily differentiable.
The symmetries of complex molecular structures can be modeled by the {\em topological symmetry group} of the underlying embedded graph. It is therefore important to understand which topological symmetry groups can be realized by particular abstract graphs. This question has been answered for complete graphs; it is natu…
Gradient estimates, Bakry-Emery Ricci curvature and ellipticity for unbounded graph Laplaciansmath.DG
Gradient estimates for unbounded graph Laplacians under Bakry-Emery curvature.
problem Gradient estimates for unbounded graph Laplacians.
method Proving gradient estimates under Bakry-Emery curvature bounds for unbounded graph Laplacians with ellipticity assumption.
result Gradient estimates and applications to completeness and finiteness of stochastically complete graphs.
We find the minimal number of links in an embedding of any complete -partite graph on 7 vertices (including , which has at least 21 links). We give either exact values or upper and lower bounds for the minimal number of links for all complete -partite graphs on 8 vertices. We also look at larger complete bip…
Characterizes weakly linked pairs of complete graphs in 3D space.
problem Identifying pairs of complete graphs that are weakly linked.
method Algebraic characterisation and geometric analysis of linking cycles.
result Characterization of weakly linked pairs of complete graphs.
Generalizations of Conway-Gordon theorems for complete graphs with new key results.
problem Understanding intrinsic knotting in complete graphs.
method Integral lifts and square of linking numbers for complete graphs with arbitrary vertices.
result Sum of second coefficients of Conway polynomials is determined for rectilinear complete graphs.
Optimal diagram found for complete graphs with linear trees.
problem Finding optimal diagrams for complete graphs.
method Using a linear tree structure to minimize crossing numbers.
result Optimal diagrams without free hamiltonian cycles for odd .
We investigate the minimal number of links and knots in complete partite graphs. We provide exact values or bounds on the minimal number of links for all complete partite graphs with all but 4 vertices in one partition, or with 9 vertices in total. In particular, we find that the minimal number of links for …
Geometric matrix completion learns graph patterns and non-linear diffusion efficiently.
problem Efficiently learn graph patterns and non-linear diffusion from user/item graphs.
method Geometric deep learning on graphs with graph convolutional and recurrent neural networks.
result Outperforms state-of-the-art techniques on synthetic and real datasets.
We characterize which automorphisms of an arbitrary complete bipartite graph can be induced by a homeomorphism of some embedding of the graph in .
Graph auto-encoder predicts user-item interactions from graph data.
problem Matrix completion for recommender systems from graph data.
method Differentiable message passing on bipartite graphs.
result Competitive performance on collaborative filtering benchmarks.
Complete classification of links and spatial graphs with finite N-quandles.
problem Classifying links and spatial graphs with finite N-quandles.
method Extending fundamental quandle relationships to N-quandles of links and spatial graphs.
result Complete list of links and partial list of spatial graphs with finite N-quandles.
Graphs from knot types help identify unique knots.
problem Identifying knots uniquely.
method Created Reidemeister graphs from knot types and analyzed their properties.
result Graph isomorphism type is a complete knot invariant.
Simplified proof for a theorem about graphs.
problem Proving a generalized Conway--Gordon--Sachs theorem for complete graphs.
method Provided a shorter proof over integers.
result A simpler proof of the generalized theorem.
We classify graphs that are 0, 1, or 2 edges short of being complete partite graphs with respect to intrinsic linking and intrinsic knotting. In addition, we classify intrinsic knotting of graphs on 8 vertices. For graphs in these families, we verify a conjecture presented in Adams' "The Knot Book": If a vertex is remo…
This paper classifies complete translating solitons in 3D space.
problem Understanding translating solitons for mean curvature flow.
method Full classification of complete translating graphs in R^3.
result A complete classification of complete translating graphs in R^3.
Researchers compute connectivity of braid group in bipartite graph configuration space.
problem Understanding connectivity of braid group in complex configuration space.
method Analysis of topology, hidden symmetry, and literature results.
result Explicit computation of connectivity at infinity for braid group.
New method improves tensor completion for weakly-dependent spatiotemporal data.
problem Improving tensor completion for weakly-dependent data on graphs.
method Introducing -norm and Graph Laplacian penalties for low-rank tensor decomposition and completion.
result Improved performance in metro passenger flow prediction.
Complex equivalence classes found in graph homotopy.
problem Complexity of proper homotopy equivalence in graphs.
method Demonstrated Borel completeness and comeager equivalence classes.
result Complex equivalence classes exist in infinite graphs.
The paper explores the -Liouville property on graphs and its connections to stochastic completeness.
problem Investigating the -Liouville property on graphs and its implications.
method Characterization of -Liouville property in terms of Green function, equivalence with stochastic completeness, and comparison theorems based on inner-outer curvatures.
result Equivalence of -Liouville property and stochastic completeness on model graphs, and introduction of Dirichlet -Liouville property.
There has been much recent interest into those properties of a 3-manifold determined by the profinite completion of its fundamental group. In this paper we give readily computable criteria specifying precisely when two orientable graph manifold groups have isomorphic profinite completions. Our results also distinguish …
Study on modified Ricci curvature on graphs, proving rigidity and deriving formulas.
problem Understanding Ricci curvature on graphs, especially for specific graph types.
method Introduced modified Ricci curvature, established rigidity theorem, derived formulas for strongly regular graphs.
result Rigidity theorem for complete graphs and explicit formulas for strongly regular graphs.
The paper reveals a property of chromatic homology for complete graphs.
problem Understanding the chromatic homology of complete graphs.
method Introduced a combinatorial description of enhanced states and used it to analyze the homology.
result Showed a splitting property of the chromatic homology for certain graphs.
The aim of this work is studying translating graphs by mean curvature flow in $\Real^3$. We prove non-existence of complete translating graphs over bounded domains in $\Real^2$. Furthermore, we show that there are only three types of complete translating graphs in $\Real^3$; entire graphs, graphs between two vertical p…
We introduce new sufficient conditions for intrinsic knotting and linking. A graph on n vertices with at least 4n-9 edges is intrinsically linked. A graph on n vertices with at least 5n-14 edges is intrinsically knotted. We also classify graphs that are 0, 1, or 2 edges short of being complete partite graphs with respe…
Study on linking numbers in random book embeddings of complete graphs.
problem Distribution and mean of linking numbers in random book embeddings of complete graphs.
method Analyzes a family of two-component links arising from random embeddings of complete graphs, using Eulerian numbers and linear growth in mean linking number.
result Mean of squared linking number over all random embeddings is $ rac{i}{6}$, where is the number of interior edges.
Algorithm refines matrix ratings using hierarchical graph clustering.
problem Matrix completion with side information from social graphs.
method Hierarchical graph clustering followed by iterative refinement of matrix ratings.
result Achieves optimal sample complexity for matrix completion.
New graphs with maximum degree 4 found to be Ricci-flat.
problem Characterizing Ricci-flat graphs with maximum degree 4.
method Defined Ricci curvature on graphs and used previous results to find all such graphs.
result All Ricci-flat graphs with maximum degree at most 4 were determined.
Paper constructs graphs with girth four and distinct properties.
problem Characterize Ricci-flat graphs with specific girth.
method Constructs and characterizes graphs with girth four, focusing on edge-disjoint and vertex-disjoint 4-cycles.
result Characterizes all Ricci-flat graphs of girth four with vertex-disjoint 4-cycles.
Conway-Gordon proved that for every spatial complete graph on 6 vertices, the sum of the linking numbers over all of the constituent 2-component links is congruent to 1 modulo 2, and for every spatial complete graph on 7 vertices, the sum of the Arf invariants over all of the Hamiltonian knots is also congruent to 1 mo…
New guarantees for matrix completion from any deterministic sampling patterns.
problem Proving guarantees for low-rank matrix completion from non-random sampling schemes.
method Introduced a graph with observed entries as edges to analyze the performance of constrained nuclear norm minimization algorithm.
result The algorithm can successfully complete the matrix if the observation graph is well-connected and has similar node degrees.
Enhances knowledge graph completion with mixed geometry tensor factorization.
problem Capturing nuanced distributional properties in knowledge graphs.
method Combines Euclidean and hyperbolic geometries for tensor factorization.
result Improves link prediction accuracy with fewer parameters.
Study geodesic flow on graph-related nilmanifolds, finding integrable and non-integrable cases.
problem Understanding geodesic flow on specific geometric structures.
method Construction of first integrals to show complete integrability.
result Examples of integrable and non-integrable geodesic flows.
Study finds the shortest triply periodic graph spanning a cubic lattice.
problem Finding the shortest periodic graph with a fixed volume.
method Analyzes the body centred cubic lattice and the gyroid surface.
result The shortest graph is the srs network with quotient.
The study examines graphs over domains in product manifolds, revealing properties of geodesics and constant curvature.
problem Properties of graphs over domains in product manifolds.
method Analyzes minimal, translating, and CMC graphs over domains with piecewise smooth boundaries.
result Geodesic arcs in the boundary of domains for minimal and translating graphs, and constant curvature for CMC graphs.
We study the Seifert surfaces of a link by relating the embeddings of graphs by using induced graphs. As applications, we prove that every link is the boundary of an oriented surface which is obtained from a graph embedding of a complete bipartite graph , where all voltage assignments on the edges of $K_{2…
Proposes a new method for predicting missing relations in knowledge graphs.
problem Predicting missing relations between entities in knowledge graphs.
method Relational message passing method considering only edge features without entity IDs.
result PathCon method outperforms state-of-the-art methods significantly.
The paper explores linked cycles in graphs and their properties.
problem Understanding the structure of linked cycles in graphs.
method Analyzing the set of all pairs of disjoint cycles in graphs and showing conditions for minimally linked sets.
result A minimally linked set of cycles in a complete graph has at most eighteen elements.
In this thesis, we analyze the stochastic completeness of a heat kernel on graphs which is a function of three variables: a pair of vertices and a continuous time, for infinite, locally finite, connected graphs. For general graphs, a sufficient condition for stochastic completeness is given in terms of the maximum vale…
The paper studies curvature conditions on birth-death processes and graphs.
problem Curvature dimension conditions on birth-death processes and linear graphs.
method Combinatorial characterization and proof of conditions for linear graphs.
result Volume doubling property and Poincaré inequality for graphs with non-negative curvature.
Low-Rank Covariance Completion for Graph Quilting with Applications to Functional Connectivitystat.ME
New method estimates neuronal connectivity from partially observed data.
problem Estimating neuronal connectivity from partially observed data.
method Two-step approach: low-rank covariance completion followed by graph structure estimation.
result Graph selection consistency demonstrated for one approach.
Paper tackles NP-complete subgraph isomorphism counting problem.
problem Counting subgraph isomorphisms in large graphs.
method Learning framework that augments representation learning architectures and iteratively attends pattern and target graphs.
result Scalable learning approach counts subgraph isomorphisms in linear time.