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.
This study examines how removing edges from complete graphs affects Ollivier Ricci curvature.
problem Conditions under which Ollivier Ricci curvature changes sign after edge removal.
method Defined and analyzed graphs obtained by removing matching, vertex incident, and cycle edges from complete graphs.
result Ollivier Ricci curvature remains positive or zero for graphs formed by removing edges from complete graphs.
Characterizes graphs with Lin-Lu-Yau curvature at least one and explores bone-idle graphs.
problem Characterizing graphs with specific curvature properties.
method Study of Ollivier-Ricci curvature and Lin-Lu-Yau curvature, exploration of regular graphs, and exact formula derivation.
result Characterizes edges that are bone-idle in regular graphs and provides a complete characterization of 4-regular bone-idle graphs.
Curvature formulas on regular graphs identified bone idle edges and graphs.
problem Understanding curvature in regular graphs and identifying bone idle edges.
method Explicit formulas for Lin-Lu-Yau and Ollivier-Ricci curvatures derived from graph parameters.
result Equality condition on regular graphs for Ollivier-Ricci curvature and characterization of bone idle edges.
Graphs with stronger curvature grow faster.
problem Understanding volume growth on graphs with various curvatures.
method Examined inner-outer and Ricci-Ollivier curvatures to relate them to volume growth.
result Graphs with stronger inner-outer curvature growth have faster volume growth.
New curvature concept preserves graph distances under operations.
problem Preserving graph distances under graph operations.
method Characterization of distance matrix and its null space.
result Linear system Dx=1 may not have a solution. New bounds for average graph distance using curvature and centrality.
problem Finding bounds for average graph distance.
method Using weighted average Ollivier curvature with edge betweenness centrality.
result Equality in bounds achieved for specific reflective graphs.
Study classifies Halin graphs with positive curvature.
problem Classifying Halin graphs with specific curvature.
method Analyzing generalized Halin graphs formed by connecting tree leaves.
result Identified all generalized Halin graphs with positive Lin-Lu-Yau curvature.
Study classifies graphs with positive curvature without quadrilaterals.
problem Classifying graphs with positive Lin-Lu-Yau curvature without quadrilaterals.
method Definition of Ricci curvature on graphs, limit-free formulation using graph Laplacian.
result Identifies all simple connected C4-free graphs with positive Lin-Lu-Yau curvature.
We propose a new graph kernel for graph classification and comparison using Ollivier Ricci curvature. The Ricci curvature of an edge in a graph describes the connectivity in the local neighborhood. An edge in a densely connected neighborhood has positive curvature and an edge serving as a local bridge has negative curv…
The study examines discrete curvature notions on Cayley graphs of certain groups.
problem Understanding curvature in discrete settings for various groups.
method Introduced Right Angled Artin-Coxeter Hybrids (RAACHs) and derived curvatures of Cayley graphs.
result Addition of relators does not decrease weighted curvatures of Cayley graphs.
We study the Bakry-Émery curvature function KG,x:(0,∞]→R of a vertex x in a locally finite graph G systematically. Here KG,x(N) is defined as the optimal curvature lower bound K in the Bakry-Émery curvature-dimension inequality $CD(\mathcal{K},\ma…
Combines curvature descriptors with TDA for graph model evaluation.
problem Evaluating graph generative models efficiently and accurately.
method Combines graph curvature descriptors with topological data analysis.
result Robust, expressive descriptors for graph generative models.
New theorem on graph curvature thresholds and uniqueness.
problem Determining the minimum number of edges for graphs to have positive curvature.
method Analyzing graphs with specific edge counts and curvature properties.
result Optimal threshold for positive curvature and uniqueness of extremal graphs.
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.
Survey on rigidity results for graphs with prescribed mean curvature.
problem Rigidity of graphs with prescribed mean curvature.
method Analysis of mean curvature operator, maximum principles, gradient estimates.
result Detailed geometric applications, including Bernstein theorem and splitting theorem.
Two complete graphs are connected by adding some edges. The obtained graph is called the gluing graph. The more we add edges, the larger the Ricci curvature on it becomes. We calculate the Ricci curvature of each edge on the gluing graph and obtain the least number of edges that result in the gluing graph having positi…
Unified estimates for mean curvature in Lorentz-Minkowski space.
problem Estimating mean curvature for space-like and time-like graphs.
method Using gradient bounds to derive Heinz-type estimates.
result Unified vanishing theorem for mean curvature of constant mean curvature graphs.
Proves existence and uniqueness of Killing graphs with prescribed curvature.
problem Existence and uniqueness of Killing graphs with prescribed curvature.
method Proves existence and uniqueness of Killing graphs with prescribed mean curvature considering non-constant functions.
result Existence and uniqueness of Killing graphs with prescribed curvature.
New curvature defined via graph resistances leads to Ricci flow.
problem Defining curvature on graph edges for analysis.
method Introducing Ricci--Foster curvature based on effective resistances and studying Ricci flow.
result Existence of solutions to Ricci flow on short time intervals, preservation of nonnegative curvature.
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.
Combinatorial approach to α-Ricci and Lin-Lu-Yau Ricci curvatures on graphs
problem Curvature formulas for α-Ricci and Lin-Lu-Yau Ricci curvatures on graphs method Combinatorial construction of optimal transport plans and exact formulas
result Combinatorial proof of known curvature formulas
The study classifies graphs with specific curvature and maximum degree.
problem Graphs with nonnegative Ricci curvature and maximum degree constraints.
method Classification of graphs with Lin-Lu-Yau-Ollivier Ricci curvature, maximum degree ≤ 3, and diameter ≥ 6.
result Classification of graphs meeting the specified criteria.
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.
We study a modified notion of Ollivier's coarse Ricci curvature on graphs introduced by Lin, Lu, and Yau in [11]. We establish a rigidity theorem for complete graphs that shows a connected finite simple graph is complete if and only if the Ricci curvature is strictly greater than one. We then derive explicit Ricci curv…
The paper reformulates Bakry-Émery curvature on graphs using eigenvalues.
problem Analyzing curvature on weighted graphs.
method Reformulating curvature as the smallest eigenvalue of a rank one perturbation of the curvature matrix.
result The curvature function is analytic, strictly monotone increasing, and concave until a threshold, after which it is constant.
A new model for graph clustering using curvature spaces.
problem Graph clustering from a geometric perspective.
method Introducing a heterogeneous curvature space and a contrastive learning approach.
result CONGREGATE model outperforms state-of-the-art competitors.
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.
Knot theory is the study of isotopy classes of embeddings of the circle S1 into a 3-manifold, specifically R3. The Fáry-Milnor Theorem says that any curve in R3 of total curvature less than 4π is unknotted. More generally, a (finite) graph consists of a finite number of edges and vertices. Given a topologica…
New curvature tensor and matrices for connection graphs derived from Bakry-Émery curvature.
problem Deriving Buser-type bounds on eigenvalues of connection Laplacians.
method Reformulation of Bakry-Émery curvature through curvature matrices and tensor representations.
result Extension of curvature matrices to connection graphs, addressing eigenfunction challenges.
Finite graphs with specific curvature have limited harmonic functions and ends.
problem Graphs with nonnegative curvature outside a finite subset.
method Introducing discrete Gromov-Hausdorff convergence to study bounded harmonic functions.
result The space of bounded harmonic functions is finite dimensional, and the number of non-parabolic ends is finite.
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.
The paper introduces a new type of Ricci flow on graphs to study their curvature.
problem Understanding the curvature of graphs and their convergence properties.
method Proposes a weighted Forman and Lin-Lu-Yau Ricci flow on graphs and proves the existence and uniqueness of solutions.
result The normalized curvature flow on trees converges to a constant curvature metric.
Spectro-Riemannian Graph Neural Networks integrate spectral and curvature signals for better graph representation learning.
problem Enhance graph representation learning by leveraging spectral and curvature signals.
method Proposes Spectro-Riemannian Graph Neural Networks (CUSP) that combines spectral and curvature insights.
result Empirical evaluation shows CUSP outperforms state-of-the-art models by up to 5.3%.
New scalar curvature defined from Ollivier-Ricci curvature for graphs.
problem Defining scalar curvature for graphs and point clouds.
method Defining a new scalar version of Ollivier-Ricci curvature and proving its convergence.
result The new scalar curvature converges to scalar curvature for sampled manifolds.
New examples of mixed-type zero-curvature graphs found.
problem Finding new examples of zero-curvature graphs in Lorentz-Minkowski space.
method Using Konderak's representation formula to construct entire zero-curvature graphs over specific planes.
result Existence of new types of entire zero-curvature graphs in mixed-type in Lorentz-Minkowski space.
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.
Study on Lin-Lu-Yau curvature and diameter of amply regular graphs.
problem Lower bounds of Lin-Lu-Yau curvature in amply regular graphs.
method Application of Hall's marriage theorem and geometric proof.
result Conference graphs have positive Lin-Lu-Yau curvature.
The study provides interior curvature estimates for convex graphs satisfying a specific quotient equation.
problem Interior curvature estimates for convex graphs.
method Analyzes convex graphs satisfying the quotient equation σn−2σn(λ)=f(X)>0. result Interior curvature estimates for convex graphs.
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.
Intertwining curvature bounds for graphs and quantum Markov semigroups verified.
problem Intertwining curvature bounds for graphs and quantum Markov semigroups.
method Introducing and verifying curvature bounds in various examples.
result Improved entropic curvature bounds for depolarizing semigroups and qubits.
On one hand, we study the class of graphs on surfaces, satisfying tessellation properties, with positive Forman curvature on each edge. Via medial graphs, we provide a new proof for the finiteness of the class, and give a complete classification. On the other hand, we classify the class of graphs on surfaces with posit…
In this paper, we study curvature dimension conditions on birth-death processes which correspond to linear graphs, i.e., weighted graphs supported on the infinite line or the half line. We give a combinatorial characterization of Bakry and Émery's CD(K,n) condition for linear graphs and prove the triviality of edge w…
We introduce and study the conical curvature-dimension condition, CCD(K,N), for graphs. We show that CCD(K,N) 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.…
The paper introduces curvature-based clustering algorithms for graph analysis.
problem Identifying densely connected substructures in graphs for community detection.
method Discrete Ricci curvatures and geometric flows to reveal community structure.
result The curvature-based approach can identify overlapping communities in graphs.
The paper estimates Betti numbers for graphs with specific curvatures, proving bounds and characterizing rigidity.
problem Estimating Betti numbers for graphs with non-negative curvatures.
method Establishing Betti number estimates for graphs with non-negative Ollivier and Bakry-Émery curvatures.
result Upper bounds on the first Betti number for graphs with non-negative curvatures, with characterizations of rigidity.
We study the evolution of convex complete non-compact graphs by positive powers of Gauss curvature. We show that if the initial complete graph has a local uniform convexity, then the graph evolves by any positive power of Gauss curvature for all time. In particular, the initial graph is not necessarily differentiable.
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.