We introduce a topological invariant, it a type of a graph-manifold, which takes natural values. For a 4-dimensional graph-manifold, whose type does not exceed two, it is proved that its universal cover is bi-Lipschitz equivalent to a universal cover of an orthogonal graph-manifold (for any Riemannian metrics on graph-…
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48 results for “graph manifold”
Graphons connect graph structures to manifold properties.
problem Interpolating between graphs and manifolds.
method Graph-to-graphon and graphon-to-manifold convergence.
result Established monotonicity inequality linking combinatorial and geometric parameters.
Proves Singer conjecture for graph manifolds with residually finite groups.
problem Proving the Singer conjecture for graph manifolds with specific properties.
method Used residual finiteness and graph manifold properties to prove the conjecture.
result Proved the Singer conjecture for extended graph manifolds and pure complex-hyperbolic higher graph manifolds.
The graph complexity of a compact 3-manifold is defined as the minimum order among all 4-colored graphs representing it. Exact calculations of graph complexity have been already performed, through tabulations, for closed orientable manifolds (up to graph complexity 32) and for compact orientable 3-manifolds with toric …
The paper explores representations of graph manifolds to Seifert motion groups.
problem Existence of faithful representations of graph manifolds to Seifert motion groups.
method Discussion and proof of non-existence of certain representations.
result Graph manifolds can have virtually no faithful representations to the Seifert motion group.
The study provides a structure theorem for a new class of noncompact 3-manifolds.
problem Understanding a new class of noncompact 3-manifolds.
method Proved a structure theorem for irreducible open graph manifolds.
result A canonical 'reduced' decomposition of irreducible open graph manifolds along embedded, incompressible 2-tori.
No Einstein metrics found on extended graph 4-manifolds.
problem Finding Einstein metrics on extended graph 4-manifolds.
method Defined and analyzed extended graph 4-manifolds as per [FLS15].
result Extended graph 4-manifolds do not support Einstein metrics.
This is an exposition of results on the existence problem of -injective immersed and embedded surfaces in graph-manifolds, and also of nonpositively curved metrics on graph-manifolds, obtained by different authors. The results are represented from a unified point of view based on the notion of compatible cohomolog…
Contact surgery graphsmath.GT
Graphs describe contact surgery on 3-manifolds.
problem Understanding contact surgery on 3-manifolds.
method Defined contact surgery graphs to analyze their properties.
result Analyzed basic properties and interesting subgraphs of contact surgery graphs.
Proposes graph neural network layers for manifold-valued graphs.
problem Graphs with features in a Riemannian manifold.
method Diffusion layer and tangent multilayer perceptron.
result Outperforms state-of-the-art networks on Alzheimer's classification.
Classifies graph configuration spaces homeomorphic to manifolds.
problem Classifying graph configuration spaces homeomorphic to manifolds.
method Developed techniques to translate topological properties into graph theoretic ones.
result Extended Abrams' work to classify certain graph configuration spaces.
We calculate the intersection ring of three-dimensional graph manifolds with rational coefficients and give an algebraic characterization of these rings when the manifold's underlying graph is a tree. We are able to use this characterization to show that the intersection ring obstructs arbitrary three-manifolds from be…
For given closed orientable 3-manifolds and let be the set of mapping degrees from to . We address the problem: For which , is finite for all ? The answer is known in Thurston's picture of closed orientable irreducible 3-manifolds unless the target is a non-trivial graph manifol…
The paper classifies when certain graph braid groups are 3-manifold groups.
problem Identifying when graph braid groups are 3-manifold groups.
method Analyzing the graph braid groups for specific graphs .
result The paper shows that is a 3-manifold group, but is not quasi-isometric to a 3-manifold group for .
This paper reconstructs complex graph signals using kernel methods on manifolds.
problem Reconstructing complex graph signals from samples on graph vertices.
method Kernel methods on complex manifolds, embedding vertices into higher-dimensional spaces.
result Effective reconstruction of complex graph signals, outperforming conventional methods.
The paper introduces heterogeneous manifolds for better graph embeddings.
problem Graph embeddings in Euclidean spaces often fail to capture the curvature of real-world graphs.
method The authors propose heterogeneous rotationally-symmetric manifolds with a radial dimension to account for varying curvature.
result The method improves graph embeddings by better preserving high-order structures and heterogeneous random graphs.
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 define the class of high dimensional graph manifolds. These are compact smooth manifolds supporting a decomposition into finitely many pieces, each of which is diffeomorphic to the product of a torus with a finite volume hyperbolic manifold with toric cusps. The various pieces are attached together via affine maps o…
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.
Graph semi-supervised learning classifies points on manifold using variational autoencoders and GNN.
problem Classifying points on low-dimensional manifolds using limited labeled data.
method Model data as points on a manifold, approximate manifold with VAE, construct geometric graph, solve semi-supervised node classification with GNN.
result Generalization gap diminishes with graph size and training procedure, vanishing asymptotically.
The paper characterizes graph manifolds using fold maps and embeddability of polyhedra.
problem Understanding the global topologies of graph manifolds.
method Using fold maps into the plane and embeddability of polyhedra in 3-manifolds.
result Characterizes graph manifolds via fold maps and polyhedra embeddability.
Graph manifolds with small homology have non-trivial SU(2) representations.
problem Characterizing graph manifolds with small homology.
method Topological methods avoiding gauge theory.
result Existence of irreducible SU(2) representations for certain graph manifolds.
There has been much recent interest into those properties of a 3-manifold determined by the profinite completion of its fundamental group. In this paper we give readily computable criteria specifying precisely when two orientable graph manifold groups have isomorphic profinite completions. Our results also distinguish …
Smooth manifolds can be triangulated with graphs of bounded twin-width.
problem Understanding the structure of triangulations of smooth manifolds.
method Using Whitney's triangulation method and bounding the twin-width of specific graphs.
result Compact smooth manifolds have triangulations with graphs of bounded twin-width.
The chromatic number of sphere graphs in 3-manifolds is bounded.
problem Understanding the chromatic number of sphere graphs in 3-manifolds.
method Analogous to curve graphs of surfaces, bounds are provided using the prime decomposition of 3-manifolds.
result Upper and lower bounds for the chromatic number of sphere graphs in 3-manifolds are derived.
Proves graph 3-manifold groups have two specific properties.
problem Understanding fundamental groups of graph 3-manifolds.
method Constructing sequences of covers to prove properties.
result Graph 3-manifold groups are virtually poly-free and in Lex family.
Graph manifolds are manifolds that decompose along tori into pieces with a tame -structure. In this paper, we prove that the simplicial volume of graph manifolds (which is known to be zero) can be approximated by integral simplicial volumes of their finite coverings. This gives a uniform proof of the vanishing of …
Graph coloring manifoldsmath.CO
We introduce a new and rich class of graph coloring manifolds via the Hom complex construction of Lovasz. The class comprises examples of Stiefel manifolds, series of spheres and products of spheres, cubical surfaces, as well as examples of Seifert manifolds. Asymptotically, graph coloring manifolds provide examples of…
The paper proves a stability result for translating space-like graphs in Lorentz manifolds.
problem Investigating stability of translating space-like graphs in Lorentz manifolds.
method Analyzing space-like graphs over a domain in Lorentz manifold with a specific metric and proving stability under conformal transformation.
result An interesting stability result for translating space-like graphs in is proven.
A standard fact about two incompressible surfaces in an irreducible 3-manifold is that one can move one of them by isotopy so that their intersection becomes -injective. By extending it on the maps of some 3-dimensional -manifolds into 4-manifolds, we prove that any homotopy equivalence of 4-dimensio…
Researchers found two types of graphs for 6D torus manifolds with Euler number 6.
problem Identifying and constructing 6D almost complex torus manifolds with specific Euler numbers.
method Examined labeled directed graphs associated with fixed points and isotropy spheres, used to construct manifolds and determine Chern numbers.
result Proved the existence of two types of 6D almost complex torus manifolds with Euler number 6.
Introduces a new manifold from a graph subgraph.
problem None explicitly stated in the abstract.
method Defined geometrically from a combinatorial subgraph of the Hoffman-Singleton graph.
result Geometric properties of the new manifold presented.
MFCNs use sparse graphs to approximate manifold convergence.
problem Understanding manifold neural networks (MNNs).
method Sparse graph approximation for manifold convergence.
result Method converges to continuum limit as data points increase.
New method separates graph structure from node attributes to recover lost signal.
problem Standard representation learning on attributed graphs merges incompatible metric spaces, leading to geometrically flawed alignment.
method Custom variational autoencoder that separates manifold learning from structural alignment.
result Transforms geometric conflict into interpretable structural descriptor, uncovering connectivity patterns and anomalies.
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.
Study graph-based algorithms for multi-manifold clustering with sufficient conditions.
problem Clustering data from a union of manifolds with different dimensions and intersections.
method Investigate sufficient conditions for similarity graphs to capture geometric information.
result High probability error bounds for spectral approximation of tensorized Laplacian.
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.
Curvature regularization prevents distortion in graph embeddings.
problem Graph topology patterns distort in Euclidean space, making detection difficult.
method Proposes curvature regularization to enforce flatness in embedding manifolds.
result Significant improvements in five embedding methods on open graph datasets.
The paper characterizes hyperbolic manifolds and graphs verifying a specific isoperimetric inequality.
problem Understanding the relationship between hyperbolicity and isoperimetric inequalities in manifolds and graphs.
method Characterization of hyperbolic manifolds and graphs with isoperimetric inequality, using Gromov boundary.
result Having a pole is a necessary condition for verifying the isoperimetric inequality, which can be removed.
The paper improves GNN generalization theory by considering graph manifolds.
problem Improper GNN generalization bounds ignoring graph structures.
method Taking a manifold perspective, the paper establishes GNN generalization theory.
result GNN generalization bounds decrease linearly with graph size and spectral continuity.
The paper extends previous work on Reeb graphs of smooth functions on 3D manifolds to non-orientable cases.
problem Extending the understanding of Reeb graphs to non-orientable 3D manifolds.
method Constructing smooth functions on non-orientable 3D manifolds whose Reeb graphs match prescribed graphs.
result Explicit construction of smooth functions on non-orientable 3D manifolds with prescribed Reeb graphs.
Constructs graph manifolds with many Anosov flows.
problem Finding graph manifolds supporting multiple Anosov flows.
method Cutting geodesic flows, pulling back to finite covers, and gluing compatible pairs of flows.
result Constructs graph manifolds with at least n Anosov flows for any n.
A compact 4-dimensional manifold is a non-singular graph-manifold if it can be obtained by the glueing T^2-bundles over compact surfaces (with boundary) of negative Euler characteristics. If none of glueing diffeomorphisms respect the bundle structures, the graph-structure is called reduced. We prove that any homotopy …
I prove that the spectrum of the Laplace-Beltrami operator with the Neumann boundary condition on a compact Riemannian manifold with boundary admits a fast approximation by the spectra of suitable graph Laplacians on proximity graphs on the manifold, and similar graph approximation works for metric-measure spaces glued…
Study of mean curvature flows on graphs in warped product manifolds, focusing on behavior at infinity.
problem Behavior of mean curvature flows on graphs in warped product manifolds, especially at infinity.
method Analysis of curve shortening flow and mean curvature flow on geodesic graphs for various warping functions.
result Long-time existence of mean curvature flows and vanishing of curvature and derivatives at infinity.
Classifies SU(2)-abelian graph manifolds with a single JSJ torus.
problem Classifying SU(2)-abelian 3-manifolds.
method Using Seifert coefficients and Heegaard Floer homology.
result SU(2)-abelian graph manifolds are Heegaard Floer homology L-spaces.
Eigenvalues of manifolds with cylindrical boundaries approximated by graph Laplacians.
problem Approximating eigenvalues of manifolds with cylindrical boundaries.
method Using truncated graph Laplacians constructed from -proximity graphs.
result Eigenvalues of truncated graph Laplacians converge to Dirichlet eigenvalues of the Laplace-Beltrami operator.
The paper classifies capillary graphs on manifolds with Ricci lower bounds.
problem Understanding capillary graphs on manifolds with Ricci lower bounds.
method Gradient estimate for positive CMC graphs on manifolds with Ricci lower bounds.
result Classification of capillary graphs over specific domains.