A new hypergraph expansion method treats vertices and hyperedges equally, improving node classification.
problem Information loss in hypergraph expansions on either vertex or hyperedge level.
method Proposes a new hypergraph formulation named line expansion (LE) that treats vertices and hyperedges symmetrically.
result The proposed line expansion method outperforms state-of-the-art baselines on five hypergraph datasets.
A single-vertex origami is a piece of paper with straight-line rays called creases emanating from a fold vertex placed in its interior or on its boundary. The Single-Vertex Origami Flattening problem asks whether it is always possible to reconfigure the creased paper from any configuration compatible with the metric, t…
New algorithm finds corrupted vertices in graphs with few queries.
problem Adversarial tampering of graph edges and vertices.
method Active learning algorithm with polynomial query complexity.
result Efficiently recovers corrupted vertices with small query complexity.
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…
RSHT algorithm simplifies complex shapes to points.
problem Simplifying complex shapes to points in higher dimensions.
method Combines simplicial collapses and expansions.
result Reduces triangulated d-manifolds to points using RSHT.
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 …
Computes link invariants in real projective 3-space using topological vertex.
problem Computing link invariants in RP3. method Uses geometric transition and toric Calabi-Yau threefold.
result Findings are series in Kahler parameters with positivity property.
New method extends knot theory to non-bipartite knots, revealing PDs.
problem Extending knot theory to non-bipartite knots.
method Developed a new positive decomposition (PD) for HOMFLY polynomials of non-bipartite knots.
result PD exists for non-bipartite knots, not just bipartite ones.
We extend average edge order results to normal 3-pseudomanifolds.
problem Determining the average edge order of normal 3-pseudomanifolds.
method Extending previous results on 3-manifolds to 3-pseudomanifolds with singularities.
result For a normal 3-pseudomanifold K, μ0(K)≥730, with equality if and only if K is a specific triangulation of RP2. The study characterizes 3-pseudomanifolds with up to two singularities.
problem Characterizing face-number-related invariants of normal 3-pseudomanifolds with up to two singularities.
method Proves properties of normal 3-pseudomanifolds using specific operations and upper bounds.
result Proves that normal 3-pseudomanifolds with up to two singularities are constructed from boundary complexes of 4-simplices.
This paper studies how adding leaves to a tree affects its spectral properties.
problem Investigating the asymptotic behavior of tree spectra under leaf attachment.
method Analyzing the Ricci matrix and its largest eigenvalue for trees with pendant edges added.
result The sequence of largest eigenvalues converges to a limit that depends on local branch data.
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 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.
Given a vertex of interest in a network G1, the vertex nomination problem seeks to find the corresponding vertex of interest (if it exists) in a second network G2. A vertex nomination scheme produces a list of the vertices in G2, ranked according to how likely they are judged to be the corresponding vertex of …
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.
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.
Quasi-vertex-transitive maps are the homogeneous maps on the plane with finitely many vertex orbits under the action of their automorphism groups. We show that there exist quasi-vertex-transitive maps of types [p3,3] for p≡1 (mod 6), but there doesn't exist vertex-transitive map of such types. In particu…
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…
New hypergraph neural network learns variable-sized hyperedges.
problem Learning representations for non-uniform hypergraphs with variable cardinalities.
method Developed a hypergraph neural network exploiting incidence structure.
result Significant improvement in accuracy on real-world hypergraph datasets.
We prove that there exists a geodesic trajectory on the dodecahedron from a vertex to itself that does not pass through any other vertex.
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.
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.
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.
A semi-regular tiling of the hyperbolic plane is a tessellation by regular geodesic polygons with the property that each vertex has the same vertex-type, which is a cyclic tuple of integers that determine the number of sides of the polygons surrounding the vertex. We determine combinatorial criteria for the existence, …
In this paper, we develop a new aligned vertex convolutional network model to learn multi-scale local-level vertex features for graph classification. Our idea is to transform the graphs of arbitrary sizes into fixed-sized aligned vertex grid structures, and define a new vertex convolution operation by adopting a set of…
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.
Given a graph in which a few vertices are deemed interesting a priori, the vertex nomination task is to order the remaining vertices into a nomination list such that there is a concentration of interesting vertices at the top of the list. Previous work has yielded several approaches to this problem, with theoretical re…
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…
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.
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.
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.
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…
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…
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.
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…
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.
While many multiple graph inference methodologies operate under the implicit assumption that an explicit vertex correspondence is known across the vertex sets of the graphs, in practice these correspondences may only be partially or errorfully known. Herein, we provide an information theoretic foundation for understand…
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 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…