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.
A vertex-transitive map X is a map on a closed surface on which the automorphism group Aut(X) acts transitively on the set of vertices. If the face-cycles at all the vertices in a map are of same type then the map is said to be a semi-equivelar map. Clearly, a vertex-transitive map is semi-equivelar. Converse…
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, …
Research tackles learning vertex representations for bipartite networks.
problem Lack of research on learning vertex representations for bipartite networks.
method Apply generic methods like node2vec and LINE, but ignore vertex type information.
result Generic methods are suboptimal for bipartite networks due to different properties and patterns.
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.
Every link in R^3 can be represented by a one-vertex ribbon graph. We prove a Markov type theorem on this subset of link diagrams.
Study integrability of quantized six-vertex model on torus.
problem Integrability of a specific lattice model on a torus.
method Defined layer transfer matrices and tetrahedron equations for admissible graphs.
result Established commutativity of transfer matrices and derived quantum Hamiltonians.
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.
Artin groups of type Dn have special cycles and complexes with interesting properties.
problem Characterizing cycles and complexes in Artin groups of type Dn. method Analyzing 6-cycles and their centers/quasi-centers in the 1-skeleton of the Artin complex.
result Certain 6-cycles in the Artin complex of type Dn have centers or quasi-centers. Study meromorphic open-string vertex algebras and modules over Riemannian manifolds.
problem Characterize meromorphic open-string vertex algebras and their modules over Riemannian manifolds.
method Explicitly determine bases for meromorphic open-string vertex algebras and their modules, using parallel tensors and eigenfunctions of the Laplace-Beltrami operator.
result Every irreducible module of a specific type is completely reducible if every composition factor is generated by eigenfunctions of eigenvalue p(p−1)K for some p∈Z+. Graph Lie algebras have infinite prolongation if they have a vertex of degree one.
problem Determining when the prolongation of a graph Lie algebra is infinite-dimensional.
method Analyzing labeled direct graphs and their associated Lie algebras.
result Graph Lie algebras are infinite-dimensional if and only if they have a vertex of degree one.
We develop an algorithm of polynomial time complexity to construct the Grushko decomposition of fundamental groups of graphs of free groups with cyclic edge groups. Our methods rely on analysing vertex links of certain CAT(0) square complexes naturally associated with a special class of the above groups. Our main resul…
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.
New method for learning on heterogeneous graphs without meta-paths.
problem Learning on heterogeneous graphs is sensitive to meta-paths choice, leading to poor performance.
method Decompose heterogeneous graph into homogeneous relation-type graphs, combine higher-order representations, use attention mechanisms.
result Our model outperforms state-of-the-art baselines in vertex classification tasks on heterogeneous graph datasets.
If the face-cycles at all the vertices in a map on a surface are of same type then the map is called semi-equivelar. There are eleven types of Archimedean tilings on the plane. All the Archimedean tilings are semi-equivelar maps. If a map X on the torus is a quotient of an Archimedean tiling on the plane then the map…
A new method pools graph vertex representations into a function space for better graph classification.
problem Pooling graph vertex representations for graph-centric tasks.
method Proposes a novel function space pooling method for graph neural networks.
result The proposed method generally outperforms existing pooling methods in graph classification.
A piecewise flat Finsler metric on a triangulated surface M is a metric whose restriction to any triangle is a flat triangle in some Minkowski space with straight edges. One of the main purposes of this work is to study the properties of geodesics on a piecewise flat Finsler surface, especially when it meets a vertex…
Semi-Equivelar maps are generalizations of Archimedean solids to the surfaces other than 2-sphere. In earlier work a complete classification of semi-equivelar map of type (35,4) on the surface of Euler characteristic -1 was given. In the meantime Karabas an Nedela classified vertex transitive semi-equivelar maps on…
A vertex-transitive map X is a map on a surface on which the automorphism group of X acts transitively on the set of vertices of X. If the face-cycles at all the vertices in a map are of same type then the map is called a semi-equivelar map. Clearly, a vertex-transitive map is semi-equivelar. Converse of this is …
The paper introduces a quantum state system to count perfect matchings in graphs.
problem Counting perfect matchings in graphs using quantum state systems.
method Topological quantum field theory (TQFT) and spectral sequences.
result The filtered n-color vertex homology for n=2 is generated by perfect matchings. Pyramidal GNN combines RC and pooling for efficient graph embeddings.
problem Efficiently embedding graphs while maintaining accuracy.
method Alternates RC layers with pooling to reduce complexity.
result Formally shows how pooling reduces complexity and speeds convergence.
We prove diameter bounds for graphs having positive Ricci-curvature bound in Bakry-Emery sense. One result using only curvature and maximal vertex degree is sharp in case of hypercubes. The other result depends on an additional dimension bound, but is independent of the vertex degree. In particular, the second result i…
We study the {\it arc and curve} complex AC(S) of an oriented connected surface S of finite type with punctures. We show that if the surface is not a sphere with one, two or three punctures nor a torus with one puncture, then the simplicial automorphism group of AC(S) coincides with the natural image of the exten…
The study proves spherical polygon analogs of curve theorems, finding bounds on intersections and inflections.
problem Finding bounds on intersections and inflections for spherical polygons.
method Adapting smooth curve theorems to spherical polygons using discrete tools.
result Proves discrete analogs of four-vertex theorems for spherical polygons.
The paper studies graph products of groups and recovers graph and vertex groups under certain conditions.
problem Recovering graph and vertex groups from graph products of groups.
method Using non-generic almost positive sentences, the authors show that under specific conditions, the underlying graph and vertex groups can be recovered.
result The core of the defining graph determines an invariant of the elementary theory of a right-angled Artin group.
We classify topologically trivial Legendrian Θ-graphs and identify the complete family of nondestabilizeable Legendrian realizations in this topological class. In contrast to all known results for Legendrian knots, this is an infinite family of graphs. We also show that any planar graph that contains a subdivision of…
The main ob jective of this research is to find the different types of elliptic triangulations for planar discs and spheres. We begin in Chapter 1 with the mandatory introduction. In the second chapter we define and study the notion of a patch, that is, a triangulation of a planar disc. By introducing a suitable notion…
Method samples triangulations of manifolds using biased random walks.
problem Efficiently sample triangulations of manifolds.
method Biased random walk through Pachner graph with Metropolis-Hastings accept/reject probabilities.
result Samples triangulations at random from chosen probability, estimating rare triangulations.
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.
This paper sets thresholds for recovering vertex correspondences in partially correlated graphs.
problem Recovering hidden vertex correspondences in partially correlated graphs.
method Proposed partially correlated Erdős-Rényi graphs model; information-theoretic thresholds; correlated functional digraphs.
result Optimal rates for partial and exact recovery of vertex correspondences.
We prove that the torsion of any closed space curve which bounds a simply connected locally convex surface vanishes at least 4 times. This answers a question of Rosenberg related to a problem of Yau on characterizing the boundary of positively curved disks in Euclidean space. Furthermore, our result generalizes the 4 v…
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 …
Paper proves ribbonlength grows linearly with knot complexity.
problem Proving ribbonlength grows linearly with knot complexity.
method Binary grid diagrams and bisected vertex leveling techniques.
result Ribbonlength is bounded by a linear function of the crossing number.
Study rigidity and volume optimization of hyperbolic polyhedra.
problem Rigidity and volume optimization of hyperbolic polyhedra.
method Analyzing decorated 1-3 type hyperbolic polyhedra and their metrics.
result Decorated 1-3 type hyperbolic polyhedra are rigid up to isometry and change of decorations.
We show how Ramond free neutral Fermi fields lead to a τ-function theory of BKP type which describes iso-orthogonal deformations of systems of ortogonal curvilinear coordinates. We also provide a vertex operator representation for the classical Ribaucour transformation.
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.
Knot lattice homology invariant is preserved under certain 3-manifold diffeomorphisms.
problem Preserving knot lattice homology invariants under 3-manifold diffeomorphisms.
method Examined filtered lattice chain homotopy types of negative-definite forests with one unframed vertex.
result Filtered lattice chain homotopy type is an invariant of the diffeomorphism type of resulting 3-manifolds.
We use the concept of intrinsic metrics to give a new definition for an isoperimetric constant of a graph. We use this novel isoperimetric constant to prove a Cheeger-type estimate for the bottom of the spectrum which is nontrivial even if the vertex degrees are unbounded.
This work develops a model to distinguish network and covariate information.
problem Identifying unique network and covariate information.
method Low-rank model with two-step estimation: spectral method followed by refinement.
result The method accurately recovers joint and individual components.
We study circle packings with the combinatorics of a triangulated disk in the plane and parametrize deformations of circle packings in terms of vertex rotation and cross ratios. We show that there is a Weierstrass representation formula relating infinitesimal deformations of circle packings to discrete minimal surfaces…
Study Type C skein modules using Sp(2n) webs and construct transparent elements.
problem Understanding Type C skein modules and constructing transparent elements. method Diagrammatic approach using multivariable Chebyshev polynomials and explicit braiding formulas.
result Construction of transparent elements in the skein module at roots of unity.
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.
Researchers clarify modular group representations and vertex operator algebras for 3d invariants.
problem Understanding the full set of 3d invariants and their modular properties.
method Introducing supersymmetric defects and constructing cone vertex operator algebras.
result The full vector-valued quantum modular form for \(\widetilde{
m SL}_2(\mathbb{Z})\) captures all \(\hat Z\)-invariants of a given three-manifold.
Approximates cycles in planar and bounded-genus graphs.
problem Finding many disjoint cycles in planar and bounded-genus graphs.
method Constant-factor approximation algorithms for vertex-disjoint and edge-disjoint cycles.
result First algorithms for vertex-disjoint paths in fully planar and bounded-genus instances.
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.