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.
Recovering core nodes from graph data with missing fringe interactions.
problem Recovering the core set from graph data with missing fringe interactions.
method Developed a theoretical framework and algorithms based on fixed-parameter tractability.
result Our algorithms outperform existing methods on various real-world datasets.
Small covers were introduced by Davis and Januszkiewicz in 1991. We introduce the notion of equilibrium triangulations for small covers. We study equilibrium and vertex minimal Z22-equivariant triangulations of 2-dimensional small covers. We discuss vertex minimal equilibrium triangulations of $\mathbb{R…
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.
A Seifert surgery is an integral surgery on a knot in S^3 producing a Seifert fiber space which may contain an exceptional fiber of index 0. The Seifert Surgery Network is a 1-dimensional complex whose vertices correspond to Seifert surgeries; its edges correspond to single twistings along "seiferters" or "annular pair…
We give three constructions of a vertex-minimal triangulation of 4-dimensional real projective space RP4. The first construction describes a 4-dimensional sphere on 32 vertices, which is a double cover of a triangulated RP4 and has a large amount of symmetry. The second and third construct…
Tollefson described a variant of normal surface theory for 3-manifolds, called Q-theory, where only the quadrilateral coordinates are used. Suppose M is a triangulated, compact, irreducible, boundary-irreducible 3-manifold. In Q-theory, if M contains an essential surface, then the projective solution space has an e…
Given a flag in each of the vertex-transitive tessellations of the Euclidean plane by regular polygons, we determine the flag stabilizer under the action of the automorphism group of a regular cover. In so doing we give a presentation of these tilings as quotients of regular (infinite) polyhedra.
The article studies crystallizations of small covers over simple polytopes and finds unique crystallizations for the n-simplex.
problem Understanding crystallizations of small covers over simple polytopes.
method Examining crystallizations of small covers over the n-simplex and prism, proving uniqueness and counting equivalence classes. result Proves uniqueness of crystallization for RPn over n-simplex and counts equivalence classes for prism. Every open Riemann surface can be triangulated with equilateral triangles.
problem The structure and triangulation of Riemann surfaces.
method Constructing a holomorphic branched covering to the Riemann sphere and glueing together equilateral triangles.
result Every open Riemann surface can be equilaterally triangulated.
Semi-Equivelar maps are generalizations of Archimedean Solids (as are equivelar maps of the Platonic solids) to the surfaces other than 2−Sphere. We classify some semi equivelar maps on surface of Euler characteristic -1 and show that none of these are vertex transitive. We establish existence of 12-covered triangula…
The study shows how nonnegative Ricci curvature and metric cones imply the existence of abelian subgroups in the fundamental group of open manifolds.
problem Understanding the structure of fundamental groups of open manifolds with specific curvature properties.
method Analyzing the properties of the Riemannian universal cover and its asymptotic cones.
result The fundamental group of an open manifold with nonnegative Ricci curvature and certain geometric properties contains an abelian subgroup of finite index.
This paper shows GNNs can learn good approximations for graph problems.
problem Learning good approximations for combinatorial graph problems.
method Developed new GNNs and bridged GNN theory with distributed local algorithms.
result Most powerful GNNs can learn approximations for minimum dominating set and vertex cover problems with specific ratios.
The study explores maps of 2- and 3-uniform tilings on the torus.
problem Understanding the number of vertex orbits in quotient maps of 2- and 3-uniform tilings.
method Analyzing the quotient maps of 2- and 3-uniform tilings on the torus.
result Bounds on the number of vertex orbits in quotient maps of 2- and 3-uniform tilings.
With the [0,1,2]-family of cyclic triangulations we introduce a rich class of vertex-transitive triangulations of surfaces. In particular, there are infinite series of cyclic q-equivelar triangulations of orientable and non-orientable surfaces for every q=3k, k≥2, and every q=3k+1, k≥3. Series of cy…
The study connects triangulated surfaces to complex projective structures and circle patterns.
problem Understanding circle patterns on complex projective tori.
method Using discrete holomorphic quadratic differentials, the approach involves cross ratio systems and Delaunay angles.
result For any triangulated torus, the projection map is a covering map with at most one branch point.
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.
Extends circle pattern theorem to quasi-simplicial triangulations.
problem Characterize circle patterns on quasi-simplicial triangulated surfaces.
method Use finite covering technique to reduce problem to simplicial case, prove characterization by KAT inequalities.
result Curvature image is characterized by KAT inequalities.
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 …
Graph dynamics link combinatorics to geometry, revealing manifold intersections and stability.
problem Understanding the geometry of graph dynamical systems with odd interactions.
method Proved geometry and stability of manifolds governed by graph homology and coverings.
result Derived upper and lower bounds on the dimension of the equilibrium set.
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.
We show that if X is an indecomposable PD3-complex and π1(X)isthefundamentalgroupofareducedfinitegraphoffinitegroupsbutisnotvirtuallycyclicthenXisorientable,theunderlyinggraphisatree,alltheedgegroupsareZ/2Zandallbutatmostoneofthevertexgroupsisdihedraloforder2m…
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.
A finite subset S of a closed hyperbolic surface F canonically determines a "centered dual decomposition" of F: a cell structure with vertex set S, geodesic edges, and 2-cells that are unions of the corresponding Delaunay polygons. Unlike a Delaunay polygon, a centered dual 2-cell Q is not determined by its collection …
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.
Seeds help match noisy graphs faster and more efficiently.
problem Matching noisy graphs with side information.
method Large neighborhood statistics for seeded graph matching.
result Achieves information-theoretic limit in polynomial time.
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…
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 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.
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.
We define metric bundles/metric graph bundles which provide a purely topological/coarse-geometric generalization of the notion of trees of metric spaces a la Bestvina-Feighn in the special case that the inclusions of the edge spaces into the vertex spaces are uniform coarsely surjective quasi-isometries. We prove the e…
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…
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.
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.
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 …
The study proves a discrete Blaschke theorem for convex polygons in 2-dimensional space forms.
problem Investigating curvature and circumradius constraints for convex polygons in 2-space forms.
method Defining curvature at each vertex and proving a Blaschke-type theorem.
result The circumradius of a convex polygon satisfies a specific inequality related to its vertex curvatures.
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.