The paper examines circle graphs of Gauss diagrams and finds counterexamples to previous descriptions.
problem Problems with previous descriptions of realizable Gauss diagrams.
method Experimental checking and formulation of new descriptions of realizable circle graphs.
result New descriptions of realizable circle graphs and an algorithm for checking realizability.
Incorrect parity-based descriptions of realizable Gauss diagrams found, but bipartite graphs provide a valid approach.
problem Incorrect descriptions of realizable Gauss diagrams using parity conditions.
method Used bipartite graphs to describe realizable Gauss diagrams.
result Realizable Gauss diagrams can be accurately described using bipartite graphs.
This paper develops graph theory for racks and quasigroups.
problem Characterizing and realizing right quasigroups and related structures.
method Study of graph markings, Schreier graphs, and Cayley graphs.
result All right quasigroups are realizable by specific types of graphs.
Groups of homotopy equivalences of graphs help realize compact subgroups.
problem Realizing compact subgroups of homotopy equivalences of graphs.
method Introduced a Polish group topology on the group of proper homotopy equivalences and proved the Nielsen Realization theorem.
result Compact subgroups of homotopy equivalences can be realized by simplicial isomorphisms of graphs.
Investigates conditions for GKM fiber bundles and realizability of fiber bundles of GKM graphs.
problem Conditions for GKM fiber bundles and realizability of fiber bundles of GKM graphs.
method Analysis of GKM graphs and fiber bundles, counterexamples, and classification of twist automorphisms.
result Realizability of fiber bundles of GKM graphs depends on the twist automorphism and can be decided in terms of the classification.
New findings on how graphs can be realized as Reeb graphs of smooth functions.
problem Realizing a given graph as the Reeb graph of a Morse function on a manifold.
method Investigates the conditions under which a graph can be realized as the Reeb graph of a Morse function on a manifold.
result For any graph with a good orientation, there exists an n-manifold and a Morse function whose Reeb graph is isomorphic to the given graph.
The paper solves graph realization problems for Reeb graphs of Morse functions.
problem Realizing graphs as Reeb graphs with specific preimage configurations.
method Constructing Morse functions with prescribed preimages.
result Solved realization problems for certain types of graphs.
Researchers prove any graph can be realized as Reeb graph, linking it to manifold properties.
problem Realizing graphs as Reeb graphs on manifolds.
method Proving graphs can be realized as Reeb graphs under natural conditions, linking Reeb number to fundamental group corank.
result Reeb number equals corank of fundamental group, extending previous results.
The paper provides conditions for realizing graphs and polytopes with specified edge lengths.
problem Proving the existence of planar embeddings or polyhedra with specified edge lengths.
method Practical sufficient conditions and software verification for non-self-intersecting perturbations of initial realizations.
result Existence of planar embeddings and polyhedra with specified edge lengths.
Riemannian manifolds can be realized as leaf spaces of matchbox manifolds.
problem Realizing Riemannian manifolds as leaf spaces of matchbox manifolds.
method Graph coloring techniques to prove realization of manifolds as leaves.
result Any repetitive Riemannian manifold of bounded geometry can be realized as a leaf of a minimal Riemannian matchbox manifold without holonomy.
We proved in another paper that every connected graph can be realized as the cut locus of some point on some riemannian surface. Here we give upper bounds on the number of such realizations.
New method realizes planar graphs as Reeb graphs of algebraic functions.
problem Realizing planar graphs as Reeb graphs of algebraic functions.
method Generic embedding and elementary procedures.
result Generically embedded planar graphs are homeomorphic to Reeb graphs of algebraic functions.
We prove that every connected graph can be realized as the cut locus of some point on some Riemannian surface S S S which, in some cases, has constant curvature. We study the stability of such realizations, and their generic behavior.
This paper classifies topological symmetry groups for Petersen family graphs.
problem Understanding symmetries of graphs embedded in 3D space.
method Examined all embeddings of Petersen family graphs in S 3 S^3 S 3 and classified their topological symmetry groups. result Identified all possible groups that can be realized as topological symmetry groups for each graph in the Petersen family.
The paper constructs minimal realizations of signed Gauss paragraphs using graph theory.
problem Constructing minimal realizations of signed Gauss paragraphs on surfaces.
method Theory of embedded graphs on oriented and compact PL-surfaces, intersection pairing of immersed PL-normal curves.
result The genus of the ambient surface can be a function of the maximum number of Carter's circles.
Optimal Euclidean structure minimizes energy in weighted toroidal graphs.
problem Finding the optimal Euclidean structure for weighted toroidal graphs.
method Minimizing Dirichlet energy over all possible Euclidean structures and realizations within a fixed homotopy class.
result The optimal Euclidean structure induces a weighted Delaunay decomposition.
Graph neural networks improve volatility forecasting by capturing spillover effects.
problem Forecasting multivariate realized volatility with spillover effects.
method Customized graph neural networks incorporating spillover effects from multi-hop neighbors.
result Modeling nonlinear spillover effects enhances forecasting accuracy, especially for short-term horizons.
New topological realization of Kontsevich graph complex for large dimensions.
problem Understanding the rational homotopy groups of Diff partial(D2k).
method Construction of a chain map from Kontsevich graph complex to rational singular chain complex.
result New elements in rational homotopy groups of BDiff partial(D2k) determined by cycles in graph complex.
New hyperbolic graph constructed from projections of free splitting graph.
problem Constructing a new hyperbolic graph from projections of free splitting graph.
method Using submanifold projections and geometric realization of free splitting graph.
result A new hyperbolic graph constructed for n ≥ 3 n\geq 3 n ≥ 3 . New method constructs smooth functions with specific Reeb graphs and preimages on 3D manifolds.
problem Construct smooth functions with prescribed Reeb graphs and preimages on 3D closed manifolds.
method Develops a new approach to realize graphs as Reeb graphs of smooth functions on 3D closed manifolds.
result Provides a best possible solution for functions on 3D closed manifolds.
Graph Signal Processing improves stock market volatility forecasting.
problem Forecasting realized volatility in a global stock market context.
method Integrating Graph Signal Processing into the HAR model.
result The proposed model outperforms HAR-type benchmarks.
The symmetries of complex molecular structures can be modeled by the {\em topological symmetry group} of the underlying embedded graph. It is therefore important to understand which topological symmetry groups can be realized by particular abstract graphs. This question has been answered for complete graphs; it is natu…
Study supports conjecture about pretzel links' homology.
problem Investigate extreme Khovanov homology of pretzel links.
method Geometric realizations of the independence simplicial complex.
result Supports conjecture that extreme Khovanov homology is torsion-free.
This paper identifies all topological symmetry groups for Heawood family graphs.
problem Understanding symmetries of spatial graphs in 3D space.
method Analyzing automorphisms of graphs embedded in S 3 S^3 S 3 . result All graphs in the Heawood family are intrinsically chiral.
The study constructs periodic surfaces using graph theory and applies cyclically branched coverings to identify their conformal type.
problem Constructing and identifying the conformal type of periodic surfaces with a given geometric structure.
method Graph theory and cyclically branched coverings.
result Explicit cone metrics on compact Riemann surfaces can be realized as the quotient of triply periodic polyhedral surfaces.
We give a combinatorial characterization of generic minimally rigid reflection frameworks. The main new idea is to study a pair of direction networks on the same graph such that one admits faithful realizations and the other has only collapsed realizations. In terms of infinitesimal rigidity, realizations of the former…
The study restricts groups in graph of groups structures.
problem Realizing groups as fundamental groups of graph of groups with restricted vertex groups.
method Analyzes restrictions on groups that can be realized and applies to manifold construction.
result Places constraints on groups that can be realized in graph of groups structures.
The paper calculates indices for families of Fredholm operators and their extensions.
problem Calculating indices for families of Fredholm operators and their extensions.
method Passing from a Fredholm operator to its graph, deforming the horizontal subspace.
result Index formulas for families of Fredholm realizations and self-adjoint extensions.
Algorithm learns graph ARMA processes for missing signal estimation.
problem Missing signal estimation in time-varying graph signals.
method Learning joint time-vertex power spectral density through convex relaxations.
result High accuracy in time-vertex signal estimation.
The paper solves the Nielsen realization problem for cyclic actions on surfaces.
problem Realizing finite order mapping classes as isometries of hyperbolic structures.
method Inductive procedure to construct hyperbolic structures and combinatorial perspective.
result Explicit solutions to the Nielsen realization problem for cyclic subgroups.
The paper defines invariants for almost graph embeddings and explores their properties.
problem Understanding the properties and limitations of almost graph embeddings in the plane.
method Introducing and analyzing integer invariants (winding number, Wu numbers) for almost embeddings.
result Some values of invariants are realizable for almost embeddings but not for embeddings.
Graph neural networks improve volatility forecasts and portfolio performance.
problem Improving volatility forecasting for better portfolio performance.
method Compared Heterogeneous Autoregressive and Long Short-Term Memory models with GraphSAGE models built on rolling correlation, sector, and Granger-causal graphs.
result GraphSAGE models with macro regime features outperform other models in terms of forecast accuracy, ranking quality, and portfolio Sharpe ratio.
Uniformizes surfaces using discrete harmonic maps and hyperbolic metrics.
problem Uniformizing surfaces with complex geometries.
method Least Dirichlet energy harmonic embedding of graphs on surfaces.
result Existence of hyperbolic metrics realizing least energy embeddings.
Graph manifolds' Thurston norms are sums of linear functionals, and every such norm can be realized.
problem Understanding Thurston norms of graph manifolds and their realizability.
method Analyzing the structure of Thurston norms as sums of linear functionals and showing realizability.
result Every Thurston norm of a graph manifold can be expressed as a sum of absolute values of linear functionals with rational coefficients.
We showed in another paper [arXiv:1103.1759] that every connected graph can be realized as the cut locus of some point on some riemannian surface S S S . Here, criteria for the orientability of S S S are given, and are applied to classify the distinct, orientable, cut locus structures on graphs with four generating cycles.
In this paper we find strictly locally convex hypersurfaces in R n + 1 \mathbb{R}^{n+1} R n + 1 with prescribed curvature and boundary. The main result is that if the given data admits a strictly locally convex radial graph as a subsolution, we can find a radial graph realizing the prescribed curvature and boundary. As an applicatio…
Graph Neural Network improves volatility forecasting for 500 S&P stocks.
problem Forecasting short-term realized volatility in a multivariate setting.
method Graph Transformer Network for Volatility Forecasting.
result Our model outperforms benchmarks on 500 S&P stocks.
This paper extends combinatorial semi-bandits to graph feedback, improving regret bounds.
problem Adversarial combinatorial semi-bandits with graph feedback.
method Introduced graph feedback in combinatorial semi-bandits, using convexified actions and online stochastic mirror descent.
result Optimal regret scales as S T + α S T S\sqrt{T}+\sqrt{αST} S T + α S T , interpolating between full and semi-bandit feedback. A simpler 3D Möbius strip design without twists.
problem Creating a Möbius strip without twists.
method A simple rational function on a polynomial subset of R^2.
result The new design is a union of a segment and a graph of a rational function.
In this work, we prove an optimal Penrose inequality for asymptotically locally hyperbolic manifolds which can be realized as graphs over Kottler space. Such inequality relies heavily on an optimal weighted Alexandrov-Fenchel inequality for the mean convex star shaped hypersurfaces in Kottler space.
This paper improves node classification using graph structure and side information.
problem Improving node classification in semi-supervised scenarios.
method Combines graph convolutional networks with extracted side information.
result The proposed model achieves higher prediction accuracy.
Small covers of graph-associahedra are URC-manifolds in higher dimensions.
problem Realizing integral homology classes as images of fundamental classes.
method Constructive proof using small covers of graph-associahedra.
result Two-fold coverings of small covers of graph-associahedra are URC-manifolds.
New results on plane graphs linked to rational functions.
problem Existence problems for plane graphs with specific degree conditions.
method Recent results on the Hurwitz existence problem.
result Description of Belyi functions corresponding to such graphs.
Reformulates RBF networks for graph-based data.
problem Applying RBF networks to graph data.
method Reformulate RBF networks for adjacency matrices, derive gradient updates.
result Guaranteed same responses as vector-based RBF networks.
We investigate Legendrian graphs in ( R 3 , ξ s t d ) (\R^3, ξ_{std}) ( R 3 , ξ s t d ) . We extend the classical invariants, Thurston-Bennequin number and rotation number to Legendrian graphs. We prove that a graph can be Legendrian realized with all its cycles Legendrian unknots with t b = − 1 tb=-1 t b = − 1 and r o t = 0 rot=0 r o t = 0 if and only if it does not contain K 4 K_4 K 4 as a mi…
Chord diagrams on circles and their intersection graphs (also known as circle graphs) have been intensively studied, and have many applications to the study of knots and knot invariants, among others. However, chord diagrams on more general graphs have not been studied, and are potentially equally valuable in the study…
In this article we give necessary and sufficient conditions for two triples of integers to be realized as the Thurston-Bennequin number and the rotation number of a Legendrian theta-graph with all cycles unknotted. We show that these invariants are not enough to determine the Legendrian class of a topologically planar …
Disk and sphere graphs embed quasi-isometrically in R^2.
problem Embedding graphs in Euclidean space.
method Disk and sphere graphs constructed from handlebodies, quasi-isometric embeddings proven.
result Quasi-isometric embeddings of disk and sphere graphs in R^2.