We introduce the notion of Bonnet-Myers and Lichnerowicz sharpness in the Ollivier Ricci curvature sense. Our main result is a classification of all self-centered Bonnet-Myers sharp graphs (hypercubes, cocktail party graphs, even-dimensional demi-cubes, Johnson graphs , the Gosset graph and suitable Cartesian …
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48 results for “Bonnet-Myers sharp graphs”
The paper calculates graph Ricci curvature and finds properties of specific graph types.
problem Understanding Ricci curvature on irregular graphs.
method Developed a formula for graph Ricci curvature based on optimal bijections.
result Derived structural and theorem results for specific graph types.
The paper refines Steinerberger curvature for block graphs and bridges.
problem Understanding curvature in graph theory.
method Formulas and relations for curvature in block graphs and graph bridges.
result Self-centered Bonnet-Myers sharp graphs are antipodal.
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 give rigidity results for the discrete Bonnet-Myers diameter bound and the Lichnerowicz eigenvalue estimate. Both inequalities are sharp if and only if the underlying graph is a hypercube. The proofs use well-known semigroup methods as well as new direct methods which translate curvature to combinatorial properties.…
Sharp spectral theorems and isoperimetric inequalities for manifolds with nonnegative Ricci curvature.
problem Understanding the geometry and topology of manifolds with nonnegative Ricci curvature.
method New spectral inequalities and isoperimetric problems involving unequal weights and warped bubbles.
result Sharp spectral and isoperimetric bounds for manifolds with nonnegative Ricci curvature.
We prove a Bonnet-Myers type theorem for quaternionic contact manifolds of dimension bigger than 7. If the manifold is complete with respect to the natural sub-Riemannian distance and satisfies a natural Ricci-type bound expressed in terms of derivatives up to the third order of the fundamental tensors, then the manifo…
Maximal diameter theorem for graphs with positive Ricci curvature.
problem Diameter comparison in directed graphs with positive Ricci curvature.
method Introduced a Lin-Lu-Yau type Ricci curvature for directed graphs and investigated rigidity properties for the equality case.
result Concluded a maximal diameter theorem of Cheng type.
The study establishes inequalities on Finsler manifolds with weighted Ricci curvature.
problem Investigating inequalities on Finsler manifolds with weighted Ricci curvature.
method Volume comparison, Bonnet-Myers theorem, Poincaré-Lichnerowicz inequality.
result Sharp lower bound for the first eigenvalue on Finsler manifolds.
The study bounds the effective diameter of graphs with positive Ollivier curvature.
problem Bounding the effective diameter of graphs with positive Ollivier curvature.
method Introducing reflective graphs and proving discrete Bonnet Myers theorem.
result The effective diameter bound is attained only for specific graphs.
New curvature measure for graphs improves diameter and eigenvalue estimates.
problem Estimating properties of graphs using Ricci curvature.
method Introduced integral Ricci curvature for graphs.
result Uniform estimates for diameter, number of vertices, and eigenvalue.
Unified LLY Ricci curvature defined for hypergraphs.
problem Defining Ricci curvature for hypergraphs.
method Unified framework for LLY Ricci curvature on hypergraphs, establishing bounds and proving properties.
result Bonnet-Myers-type theorem for hypergraphs, highlighting curvature's potential in hypergraph analysis.
A note on a Bonnet-Myers type diameter bound for graphs with positive entropic Ricci curvaturemath.PR
The study finds a diameter bound for graphs with positive entropic Ricci curvature, with optimal bounds for arithmetic mean.
problem Finding diameter bounds for graphs with positive entropic Ricci curvature.
method Using a localized gradient estimate and an equivalent definition of entropic Ricci curvature, the study derives a Bonnet-Myers type diameter bound.
result The derived diameter bound is optimal for arithmetic mean, but not for logarithmic mean.
On H-type sub-Riemannian manifolds we establish sub-Hessian and sub-Laplacian comparison theorems which are uniform for a family of approximating Riemannian metrics converging to the sub-Riemannian one. We also prove a sharp sub-Riemannian Bonnet-Myers theorem that extends to this general setting results previously pro…
We obtain sharp quantitative Laplacian upper and lower estimates under no assumption on curvatures. As a result, we derive quantitative Laplacian, area and volume comparison theorems for tubes in Riemannian and Kähler manifolds under weak integral curvature assumptions. We also give some applications, such as a general…
We define a hybrid between Ollvier and Bakry Emery curvature on graphs with dependence on a variable neighborhood. The hexagonal lattice is non-negatively curved under this new curvature notion. Bonnet-Myers diameter bounds and Lichnerowicz eigenvalue estimates follow from the standard arguments. We prove gradient esti…
The study proves timelike Ricci bounds for low regularity spacetimes using optimal transport.
problem Proving timelike Ricci bounds for spacetimes with low regularity.
method Using optimal transport to prove timelike measure-contraction property.
result Timelike curvature-dimension condition holds for metrics.
The paper extends Bonnet-Myers theorem for manifolds with nonnegative Ricci curvature.
problem Compactness and diameter estimation for manifolds with nonnegative Ricci curvature.
method General curvature conditions for estimating diameter and compactness criteria.
result Established compactness theorems for manifolds with polynomial or exponential Ricci curvature decay.
Motivated by a classical comparison result of J. C. F. Sturm we introduce a curvature-dimension condition CD(k,N) for general metric measure spaces and variable lower curvature bound k. In the case of non-zero constant lower curvature our approach coincides with the celebrated condition that was proposed by K.-T. Sturm…
The extrinsic Bonnet-Myers theorem is proven for positive Ricci curvature manifolds.
problem Understanding the structure of compact Riemannian manifolds with positive Ricci curvature.
method Establishing the extrinsic Bonnet-Myers theorem and showing almost rigidity for hypersurfaces.
result Proven the extrinsic Bonnet-Myers theorem for positive Ricci curvature manifolds and demonstrated almost rigidity for hypersurfaces.
We give a complementary generalization of the extensions of Bonnet-Myers theorem obtained by Calabi and also Cheeger-Gromov-Taylor.
In this paper, we prove the extensions of Bonnet--Myers' type theorems obtained by Calabi and Cheeger--Gromov--Taylor via Bakry--Emery Ricci curvature, which generalize the results of \cite{FG, Lim1, Wan, Wang, WW, Wu}.
New curvature measure defined for graphs, with bounds on diameter and spectral gap.
problem Defining curvature for graphs and proving its properties.
method Solving linear systems to compute curvature; applying minimax theorem.
result Graphs with positive curvature have bounded diameter and spectral gap.
Proves closure for specific spacetimes with certain conditions.
problem Proving closure for globally hyperbolic spacetimes.
method Using a Bonnet-Myers type result.
result Proves closure for spacetimes with specific conditions.
New Alexandrov-Patchwork construction for Lorentzian spaces with curvature bounds.
problem Understanding finite diameter constraints in Lorentzian geometry.
method Constructing Alexandrov-Patchwork and proving Bonnet-Myers theorem for Lorentzian spaces.
result Lorentzian spaces with curvature bounds have finite diameter.
By studying the heat semigroup, we prove Li-Yau type estimates for bounded and positive solutions of the heat equation on graphs, under the assumption of the curvature-dimension inequality , which can be consider as a notion of curvature for graphs. Furthermore, we derive that if a graph has non-negative cur…
The study compares spectral volumes of manifolds with weakly convex boundaries.
problem Establishing volume comparison theorems for manifolds with weakly convex boundaries.
method Using spectral methods and Ricci tensor eigenvalues, the study compares volumes and diameters of manifolds.
result Sharp upper bounds for the volume and diameter of manifolds with weakly convex boundaries.
This thesis explores Ollivier-Ricci curvature in graphs and manifolds, with applications to graph neural networks.
problem Understanding curvature in metric spaces and graphs.
method Combines optimal transport theory, Riemannian manifolds, and graph theory to define and analyze Ollivier-Ricci curvature.
result Extensions of Ollivier-Ricci curvature to directed graphs and applications in network science.
The paper proves properties of Lipschitz spacetimes with bounded Ricci curvature.
problem Properties of Lipschitz spacetimes with bounded Ricci curvature.
method Globally hyperbolic spacetimes with locally Lipschitz metrics and timelike Ricci curvature.
result New comparison theorems for Lipschitz spacetimes.
Study on Kähler Finsler manifolds with curvature bounds, proving theorems.
problem Understanding Kähler Finsler manifolds with curvature constraints.
method Analyzing partial parallelism of complex structure, proving theorems.
result Generalized comparison theorem for positively curved Kähler Finsler manifolds.
Sharp estimates for mean curvature flow of graphs are shown and examples are given to illustrate why these are sharp. The estimates improves earlier (non-sharp) estimates of Klaus Ecker and Gerhard Huisken.
Sub-Laplacian comparison theorems on Riemannian foliations with minimal leaves and applicationsmath.DG
The paper proves new comparison theorems for sub-Laplacian in foliations with minimal leaves.
problem Proving comparison theorems for sub-Laplacian in Riemannian foliations with minimal leaves.
method Using Riemannian foliations with minimal leaves, the paper proves comparison theorems for the sub-Laplacian.
result The comparison theorems yield a Bonnet-Myers type theorem, stochastic completeness, and Lipschitz regularization property for the sub-Riemannian semigroup.
We introduce and study the conical curvature-dimension condition, , for graphs. We show that provides necessary and sufficient conditions for the underlying graph to satisfy a sharp global Poincaré inequality which in turn translates to a sharp lower bound for the first eigenvalues of these graphs.…
For a fat sub-Riemannian structure, we introduce three canonical Ricci curvatures in the sense of Agrachev-Zelenko-Li. Under appropriate bounds we prove comparison theorems for conjugate lengths, Bonnet-Myers type results and Laplacian comparison theorems for the intrinsic sub-Laplacian. As an application, we consider …
Sharp threshold found for Frechet mean of inhomogeneous graphs.
problem Finding the Frechet mean of inhomogeneous Erdos-Renyi random graphs.
method Thresholding the expected adjacency matrix of the ensemble.
result The Frechet mean graph of inhomogeneous Erdos-Renyi random graphs exhibits a sharp threshold.
Sharp upper bound for minimal graph area in unit ball established.
problem Determining the exact upper limit for the area of minimal graphs intersecting a unit ball.
method Constructing a sequence of minimal graphs via solutions to a Dirichlet problem.
result The areas of constructed minimal graphs tend to the upper bound of .
In this note, we prove the sharp Davies-Gaffney-Grigor'yan lemma for minimal heat kernels on graphs.
Diameter theorems on Kähler and quaternionic Kähler manifolds under a positive lower curvature boundmath.DG
Study diameter bounds on Kähler and quaternionic Kähler manifolds with positive curvature.
problem Determine diameter bounds for Kähler and quaternionic Kähler manifolds under curvature positivity.
method Define orthogonal Bakry-Émery tensor, study diameter theorems, and derive Bonnet-Myers type bounds.
result Sharper diameter bounds than in Riemannian case under specific curvature assumptions.
Sharp bounds on diameter and eigenvalues for amply regular graphs.
problem Finding bounds for amply regular graphs' diameter and eigenvalues.
method New ideas relating discrete Ricci curvature to local matching properties, including a novel construction of a regular bipartite graph.
result Sharp diameter and eigenvalue bounds for amply regular graphs.
Synthetic proof shows globally hyperbolic Lorentzian spaces with specific curvature are warped products.
problem Synthetic proof of rigidity for globally hyperbolic Lorentzian spaces.
method Synthetic geometry and warped product analysis.
result Spaces with specific curvature and distance realizer are warped products.
We prove new Beckner-Sobolev type inequalities on compact Kähler manifolds with positive Ricci curvature. As an application, we obtain a diameter upper bound that improves the Bonnet-Myers bound.
Sharp thresholds and contiguity for community detection in contextual SBM.
problem Community detection in graphs with high-dimensional node-covariates.
method Contextual Stochastic Block Model, non-rigorous cavity method, information theory.
result Established the sharp threshold for detection and weak recovery in the contextual SBM.
Sharp bounds found for minimal surface solutions.
problem Finding bounds for minimal surface solutions.
method Analyzing minimal surface equation with specific boundary conditions.
result Sharp bounds established for solutions over certain domains.
Sharp curvature bounds for minimal graphs over unit disk.
problem Proving sharp curvature bounds for minimal graphs.
method Analyzing minimal graphs over unit disk, using Heinz constant and Hopf constant.
result Improved estimate for curvature of minimal graphs and sharp inequality.
We introduce a metric notion of Ricci curvature for manifolds and study its convergence properties. We also prove a fitting version of the Bonnet-Myers Theorem, for surfaces as well as for a large class of higher dimensional manifolds.
Bakry-Émery curvature sharpness and curvature flow in finite weighted graphs. II. Implementationmath.CA
Flow preserves curvature sharpness on weighted graphs.
problem Curvature flow on weighted graphs.
method Adapting Bakry-Émery calculus for Markovian preservation and analyzing limits.
result Flow limits to curvature sharp weighted graphs.
In this survey, we study three different notions of curvature that are defined on graphs, namely, combinatorial curvature, Bakry-Émery curvature, and Ollivier's Ricci curvature. For each curvature notion, the definition and its motivation from Riemannian geometry will be explained. Moreover, we bring together some glob…
Sharp bounds for spanning tree entropy in planar lattices.
problem Estimating spanning tree entropy in planar lattice graphs.
method Using hyperbolic geometry and polyhedra volumes.
result Proved bounds are easy to compute and provide excellent estimates.