Consider a group G and a family of subgroups of G. We say that vertex finiteness holds for splittings of G over if, up to isomorphism, there are only finitely many possibilities for vertex stabilizers of minimal G-trees with edge stabilizers in . We show vertex finiteness when G…
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48 results for “vertex stabilization”
Planar Legendrian -graphsmath.GT
Classifies and identifies Legendrian Θ-graphs and their embeddings.
problem Classifying and identifying Legendrian Θ-graphs and their embeddings.
method Introducing vertex stabilization and twist moves, classifying topologically trivial graphs, and identifying nondestabilizeable embeddings.
result An infinite family of nondestabilizeable Legendrian realizations in the topological class of Θ-graphs.
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.
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.
Sharp spectral gap bounds for group elements and 3-manifold groups.
problem Bounding stable commutator length in groups and 3-manifold groups.
method Characterizing maps of surfaces to K(G,1) and constructing quasimorphisms.
result Many groups have a spectral gap, with bounds depending on the group structure.
Paper solves isomorphism problem for specific Baumslag-Solitar groups.
problem Isomorphism problem for small rose non-ascending generalized Baumslag-Solitar groups.
method Analyzed group actions on trees with specific stabilizers.
result Isomorphism problem solvable for the specified groups.
A tubular group is a group that acts on a tree with vertex stabilizers and edge stabilizers. This paper develops further a criterion of Wise and determines when a tubular group acts freely on a finite dimensional CAT(0) cube complex. As a consequence we offer a unified explanation of the fai…
Scl in graphs of groupsmath.GT
Scl in groups acting on trees is rational and converges to limits.
problem Understanding stable commutator length in group actions on trees.
method Analyzing groups acting on trees with cyclic stabilizers, focusing on stable commutator length and its limits.
result Stable commutator length is rational and converges to limits in surgery families.
We study the Fibered Isomorphism conjecture of Farrell and Jones for groups acting on trees. We show that under certain conditions the conjecture is true for groups acting on trees when the stabilizers satisfy the conjecture. These conditions are satisfied in several cases of the conjecture. We prove some general resul…
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.
Explicit presentations found for asymptotically rigid mapping class groups.
problem Understanding the structure of asymptotically rigid mapping class groups.
method Using a graph of groups structure, we compute explicit presentations.
result Computed explicit presentations for asymptotically rigid mapping class groups of surfaces.
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.
In the framework of homological characterizations of relative hyperbolicity, Groves and Manning posed the question of whether a simply connected -complex with a linear homological isoperimetric inequality, a bound on the length of attaching maps of -cells and finitely many -cells adjacent to any edge must …
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 -vertex-homogeneous lattices.
method Analyzing maps as quotients of -vertex-homogeneous lattices.
result Sharp bounds of the number of vertex orbits are identified.
The paper consists of two parts. In the first one we show that a relatively hyperbolic group splits as a star graph of groups whose central vertex group is finitely generated and the other vertex groups are maximal parabolic subgroups. As a corollary we obtain that every group which admits 3-discontinuous and 2-coc…
We extend some results of [BF12] on subfactor projections to show that the projection of a free factor B to the free factor complex of the free factor A is well-defined with uniformly bound diameter, unless either A is contained in B or A and B are vertex stabilizers of a single splitting of F_n, i.e. they are disjoint…
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.
Study on combinatorial Yamabe flow on hyperbolic surfaces, proving existence and uniqueness.
problem Existence and uniqueness of solutions to combinatorial Yamabe flow on hyperbolic surfaces.
method Introduced combinatorial Yamabe flow and extended flow with generalized curvature to address potential degeneration of triangles.
result Established existence and uniqueness of solutions to the extended flow under certain conditions.
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.
Robust Vertex Classificationstat.ML
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.
This study analyzes how cryptocurrency networks adapt to financial disruptions.
problem Understanding how cryptocurrency networks respond to financial crises.
method Vertex centrality measures to assess network stability and resilience.
result Different cryptocurrencies experienced shifts in their network roles during the FTX crisis.
This work extends GNNs to handle multiple graphs with non-commuting operators, proving transferability.
problem Handling multiple graphs with non-commuting operators in graph neural networks.
method Developed a mathematical theory for graph-tuple neural networks (GtNNs) with non-commuting non-expansive operators.
result Proved universal transferability of GtNNs, ensuring no non-transferable energy under convergence.
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.
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 be a Riemannian manifold. For , the tensor algebra of the negative part of the (complex) affinization of the tangent space of at has a natural structure of a meromorphic open-string vertex algebra. These meromorphic open-string vertex algebras form a vector bundle over 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.
Marked vertex diagrams provide a combinatorial way to represent knotted surfaces in ; 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.