Vertex distortion detects if a knot is unknot.
problem Determining if a knot is the unknot.
method Using Denne-Sullivan's bound on Gromov distortion, the vertex distortion of nontrivial lattice knots is bounded. Then, it is shown that trivial vertex distortion implies the unknot.
result The conjecture that trivial vertex distortion implies the unknot is proven.
The paper extends vertex nomination schemes to general graph models and explores consistency.
problem Finding corresponding vertices in a network when given a vertex of interest.
method Extended statistical model of graphs, definitions of Bayes optimality and consistency, derivation of Bayes optimal scheme, proof of no universally consistent schemes.
result No universally consistent vertex nomination schemes exist.
The study finds the bounds of vertex orbits in maps derived from specific lattices.
problem Determining the bounds of vertex orbits in maps derived from k-vertex-homogeneous lattices. method Analyzing maps as quotients of k-vertex-homogeneous lattices. result Sharp bounds of the number of vertex orbits are identified.
New maps on the plane with specific symmetry properties identified.
problem Characterizing maps with quasi-vertex-transitive properties.
method Analyzing the automorphism groups and vertex orbits of maps on the plane.
result Existence of quasi-vertex-transitive maps of certain types, but not vertex-transitive.
634 vertex-transitive and over 10^103 non-vertex-transitive 27-vertex triangulations of octonionic projective plane.
problem Constructing and classifying triangulations of the octonionic projective plane.
method Combinatorial construction and analysis of symmetry groups.
result Found 634 vertex-transitive and over 10^103 non-vertex-transitive 27-vertex triangulations.
This paper shows semi-equivelar toroidal maps are vertex-transitive covers.
problem Understanding the relationship between semi-equivelar and vertex-transitive toroidal maps.
method Proving semi-equivelar toroidal maps are quotients of vertex-transitive toroidal maps.
result Each semi-equivelar toroidal map has a finite vertex-transitive cover.
Geodesic loop on dodecahedron avoids all vertices.
problem Existence of geodesic loop on dodecahedron.
method Proving existence of geodesic trajectory without passing through other vertices.
result Existence of a geodesic trajectory from a vertex to itself without passing through any other vertex.
Defines formal vertex laws related to Lie conformal algebras.
problem No specific problem stated; focuses on definitions and proofs.
method Definitions and proofs of vertex/conformal versions of classical Lie theory results.
result Proves vertex/conformal versions of important Lie theory results.
New relations for vertex polynomial in graphs of any degree.
problem Understanding vertex polynomial in graphs of varying degrees.
method Proved local relations for digons, triangles, quadrilaterals, and pentagons.
result Established new relations for vertex polynomial in graphs of arbitrary degree.
The study examines vertices in curves with singular points in the Euclidean plane.
problem Investigating vertices in curves with singular points in the Euclidean plane.
method Defining vertices using evolutes of frontals and analyzing conditions for the four vertex theorem.
result Conditions for the four vertex theorem to hold for closed frontals.
Vertex distortion measures how far lattice knots deviate from straight lines.
problem Measuring how much lattice knots deviate from straight paths.
method Analogous to smooth knots, study vertex distortion in lattice knots.
result Vertex distortion is 1 only for the unknot and can be arbitrarily high.
For random graphs distributed according to stochastic blockmodels, a special case of latent position graphs, adjacency spectral embedding followed by appropriate vertex classification is asymptotically Bayes optimal; but this approach requires knowledge of and critically depends on the model dimension. In this paper, w…
Paper proves ML-based vertex nomination is consistent and scalable.
problem Ordering non-interesting vertices to highlight interesting ones in graphs.
method Maximum likelihood estimation and vertex nomination scheme.
result ML-based scheme asymptotically matches Bayes optimal scheme performance.
New model learns graph features for classification.
problem Graph classification with structural information loss.
method Transform graphs into vertex grids, apply vertex convolution.
result Model preserves structural information on local vertices.
Research determines criteria for semi-regular tilings in hyperbolic space.
problem Finding combinatorial criteria for semi-regular tilings in hyperbolic geometry.
method Combinatorial analysis of vertex-types and geodesic polygons.
result Determined criteria for existence and uniqueness of semi-regular tilings.
New proof for global rigidity of vertex scaling on polyhedral surfaces.
problem Global rigidity of vertex scaling on polyhedral surfaces.
method Elementary variational proof based on continuity of eigenvalues and extension of convex functions.
result Global rigidity of vertex scaling proved without involving 3D hyperbolic geometry.
Improved accuracy in community detection with vertex labels.
problem Efficient inference in stochastic block models with vertex labels.
method Linearized belief propagation algorithm with vertex labels.
result Belief propagation achieves highest accuracy when a function of network parameters has a unique fixed point.
Proves a generalized Whitehead cut vertex lemma for tree groups.
problem Extending Whitehead's cut vertex lemma to tree group conjugacy classes.
method Proves a version of Whitehead's lemma for tree groups.
result Establishes a cut vertex in star graphs for tree group conjugacy classes.
Graph matching recovers lost vertex correspondence in shuffled graphs.
problem Errorful vertex correspondences impact graph inference performance.
method Information theory and graph matching algorithms.
result Graph matching can recover true vertex correspondence and reduce information loss.
Solves Skopenkov's problem on graph embedding criteria.
problem Criteria for toroidal embedding of one-vertex ribbon graphs.
method Analyzes one-vertex ribbon graphs with additional disc structure.
result Provides solutions to Skopenkov's problem.
Efficient method for vertex embedding and community detection.
problem Vertex embedding and community detection.
method Normalized one-hot graph encoder and rank-based cluster size measure.
result Excellent numerical performance of graph encoder ensemble algorithm.
New invariants distinguish spatial graphs not previously possible.
problem Distinguishing spatial graphs using Dehn colorings.
method Developed vertex-weight invariants based on Dehn colorings.
result Found spatial graphs distinguishable by vertex-weight invariants.
The paper explores how to find relevant vertices in one graph using another graph's attributes and structure.
problem Finding relevant vertices in one graph using another graph's attributes and structure.
method Theoretical and practical exploration of vertex nomination schemes that leverage both content (edge and vertex attributes) and context (network topology).
result Necessary and sufficient conditions for schemes that use both content and context to outperform those using only one.
The paper concerns discrete versions of the three well-known results of projective differential geometry: the four vertex theorem, the six affine vertex theorem and the Ghys theorem on four zeroes of the Schwarzian derivative. We study geometry of closed polygonal lines in $\bbRP^d$ and prove that polygons satisfying a…
Investigates the vertex curve of smooth surfaces in 3D space, connecting geometry and image analysis.
problem Understanding the geometry of smooth surfaces in 3D space.
method Analyzes the vertex curve, related to differential geometry and symmetry sets of isophote curves.
result Establishes connections between the vertex curve and other geometric curves like parabolic and flecnodal curves.
Let M be a Riemannian manifold. For p∈M, the tensor algebra of the negative part of the (complex) affinization of the tangent space of M at p has a natural structure of a meromorphic open-string vertex algebra. These meromorphic open-string vertex algebras form a vector bundle over M with a connection. We …
Proofs for Moon's theorem and its generalization.
problem Proving Moon's theorem and its generalization.
method Proofs based on key lemmas.
result Generalization of the four-vertex theorem.
Study triangulations with a unique irregular vertex of valence 6k.
problem Characterize triangulations with a specific valence for vertices.
method Use a flat singular Riemannian metric adapted to triangulations.
result Uniqueness theorem for triangulations with irregular vertex valence not a multiple of 6.
Study finds root vertex in large networks with high probability.
problem Finding the root vertex in large growing networks.
method Constructs confidence sets for the root vertex in various random network models.
result Confidence sets of size independent of the number of vertices contain the root vertex with high probability.
Consider a group G and a family A of subgroups of G. We say that vertex finiteness holds for splittings of G over A if, up to isomorphism, there are only finitely many possibilities for vertex stabilizers of minimal G-trees with edge stabilizers in A. We show vertex finiteness when G…
Marked vertex diagrams provide a combinatorial way to represent knotted surfaces in R4; including virtual crossings allows for a theory of virtual knotted surfaces and virtual cobordisms. Biquandle counting invariants are defined only for marked vertex diagrams representing knotted orientable surfaces; we e…
The study identifies 11 types of semi-equivelar maps on the torus, with some being vertex-transitive.
problem Characterizing semi-equivelar maps on the torus and identifying vertex-transitive ones.
method Analysis of automorphism groups and face-cycles.
result There are 11 types of semi-equivelar maps on the torus, with some being vertex-transitive.
Researchers link vertex algebras to non-Kähler solutions of the Hull-Strominger system.
problem Constructing representations of vertex algebras from non-Kähler solutions of the Hull-Strominger system.
method Embedding the N=2 superconformal vertex algebra in the chiral de Rham complex of a string Courant algebroid, with a condition on the Hermitian-Yang-Mills connection.
result Any solution of the Hull-Strominger system satisfying the Hermitian-Yang-Mills condition has an associated N=2 embedding.
The paper develops a spectral theory for hypergraphs with edge-dependent vertex weights using random walks.
problem Lack of spectral theory for hypergraphs with edge-dependent vertex weights.
method Random walks on hypergraphs with edge-dependent vertex weights, deriving a random walk-based hypergraph Laplacian.
result Random walks on hypergraphs with edge-dependent vertex weights can capture higher-order relationships in data.
Given a finite graph of relatively hyperbolic groups with its fundamental group relatively hyperbolic and edge groups quasi-isometrically embedded and relatively quasiconvex in vertex groups, we prove that vertex groups are relatively quasiconvex if and only if all the vertex groups have finite relative height in the f…
This paper tackles scale-free networks by preserving their heavy-tailed vertex degree distribution.
problem Preserving the scale-free property in network embeddings.
method Proposes a 'degree penalty' principle to design algorithms that preserve the heavy-tailed degree distribution of scale-free networks.
result Our algorithms reconstruct the heavy-tailed degree distribution and outperform state-of-the-art models in network mining tasks.
Let G and F be finitely generated groups with infinitely many ends and let A and B be graph of groups decompositions of F and G such that all edge groups are finite and all vertex groups have at most one end. We show that G and F are quasi-isometric if and only if every one-ended vertex group of A is quasi-isometric to…
The study proves unique harmonic functions and combinatorial properties of vertex-transitive graphs.
problem Proving combinatorial properties of vertex-transitive graphs.
method Using harmonic functions and quasi-isometry to R, proving uniqueness and combinatorial results. result Connective constant of non-degenerate vertex-transitive graphs is at least the golden mean.
Enhances graph classification with multiple graphs.
problem Improving graph classification accuracy with multiple graphs.
method Graph fusion embedding using encoder embedding.
result The method consistently improves classification accuracy for large vertex sets.
The existence of a balanced vertex is proven for geodesic nets with three boundary vertices.
problem Existence of a balanced vertex in geodesic nets with specific boundary conditions.
method Proof of existence on a general two-dimensional Riemannian surface.
result Existence of a balanced vertex for geodesic nets with three unbalanced boundary vertices.
The paper classifies and describes five-sided hyperbolic polyhedra with one ideal vertex.
problem Classifying five-sided hyperbolic polyhedra with one ideal vertex.
method Using lines and circles in the plane to find each polyhedron in the upper half-space model, and generating matrix generators for the reflection groups.
result Matrix generators for the orientation-preserving subgroup of each corresponding reflection group.
Generalizes Kauffman-Vogel polynomials to oriented and unoriented 4-valent graphs.
problem Polynomial invariants of 4-valent rigid vertex graphs.
method Using A2 bracket and A2 clasps to generalize the one-variable Kauffman-Vogel polynomial. result New polynomial invariants for oriented and unoriented 4-valent graphs.
Algorithm constructs Grushko decomposition of certain groups.
problem Decomposing fundamental groups of graphs of free groups.
method Analyzing vertex links of CAT(0) square complexes.
result Transforms complex to one with strong connectivity vertex links.
Two spectral algorithms for community detection in graphs with covariates are compared.
problem Detecting community structure in graphs with covariates.
method Two model-based spectral algorithms are presented and compared.
result The second algorithm often better estimates block assignments by accounting for vertex covariates.
Normal 4-pseudomanifolds with one or two singular vertices are derived from specific operations.
problem Understanding face-number invariants in normal 4-pseudomanifolds.
method Structural analysis and sequence of operations (vertex foldings, edge foldings, connected sums).
result Normal 4-pseudomanifolds with specific conditions can be derived from boundary complexes of 5-simplices.
We study the supersymmetric Wilson loop as introduced by Caron-Huot, which attaches to lightlike polygons certain edge and vertex operators, whose shape is determined by supersymmetry constraints. We state explicit formulas for the vertex operators to all orders in the Graßmann expansion, thus filling a gap in the lite…
We describe an algorithm for the enumeration of (candidates of) vertex-transitive combinatorial d-manifolds. With an implementation of our algorithm, we determine, up to combinatorial equivalence, all combinatorial manifolds with a vertex-transitive automorphism group on n≤13 vertices. With the exception of act…
Authors prove a formula relating the Gaussian curvature of polyhedral vertex stars to their Gauss images.
problem Proving a formula connecting discrete Gaussian curvature to the algebraic area of Gauss images.
method Comparing winding numbers and critical point index of a normal vector to deduce the formula.
result Formula significantly limits possible shapes of Gauss images of polyhedral vertex stars.