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.
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.
Finite subgraphs in flip graphs ensure unique surface embeddings.
problem Ensuring unique embeddings of surfaces based on flip graphs.
method Analyzing finite subgraphs within flip graphs of surfaces.
result Injective homomorphisms are uniquely extendable and induced by embeddings.
The paper classifies dense conjugacy classes in mapping class groups of locally finite graphs.
problem Identifying which mapping class groups have dense conjugacy classes.
method Developed flux homomorphisms and combinatorial criteria for stability.
result A complete classification for self-similar locally finite graphs and a criterion for stability.
Local existence and uniqueness of Bakry-Émery Ricci flow solutions on finite graphs.
problem Analyzing the behavior of Ricci flow on finite graphs.
method Local existence and uniqueness proof for solutions of the Bakry-Émery Ricci flow.
result Local existence and uniqueness of solutions to the Ricci flow on finite graphs.
The paper describes the K-theory of C∗-algebras of locally finite graphs.
problem Computing the K-theory of C∗-algebras of locally finite graphs. method Using a directed graph representation and Cuntz-Krieger algebra, the paper computes the K-theory of C∗(Γ). result The K-theory of C∗(Γ) is determined by the graph's genus, number of ends, and dead-ends. The sinh-Gordon equation is solved on finite, symmetric graphs.
problem Solving the sinh-Gordon equation with nonzero prescribed functions on finite graphs.
method Uniform a priori estimate to define topological degree, case-by-case calculation of degree, classical sinh-Gordon equation analysis.
result The classical sinh-Gordon equation with nonzero prescribed function is always solvable on finite, symmetric graphs.
The flip graph and arc complex of a surface are shown to have finite rigidity.
problem Finite rigidity of flip graph and arc complex for surfaces.
method Embedding the flip graph in the arc complex and leveraging finite rigidity of the flip graph.
result Finite rigidity of the flip graph implies finite rigidity of the arc complex.
Study of mapping class groups of infinite graphs, focusing on their finiteness and commensurability.
problem Understanding the finiteness properties and commensurability of mapping class groups of infinite graphs.
method Investigation of asymptotically rigid mapping class groups, construction of explicit presentations, and analysis of algebraic and geometric properties.
result Graph Houghton groups are not commensurable with other known Houghton-type groups, defining a new class of groups.
The paper develops finite knot theory using ropelength-filtered Reidemeister graphs.
problem Understanding knot types in bounded ropelength sublevel spaces.
method Study thick representatives in bounded ropelength sublevel spaces through lifted Reidemeister graphs.
result Define characteristic Reidemeister patterns and finite recognition length.
Complete classification of links and spatial graphs with finite N-quandles.
problem Classifying links and spatial graphs with finite N-quandles.
method Extending fundamental quandle relationships to N-quandles of links and spatial graphs.
result Complete list of links and partial list of spatial graphs with finite N-quandles.
Spatial graphs are decomposed into planar forests and braids.
problem Understanding the structure of spatial graphs in 3-space.
method Decomposition of spatial graphs into planar forests and braids.
result Every finite spatial graph is a connected sum of a planar graph and a braid.
In this paper, we investigate a relation between finite graphs, simplicial flag complexes and right-angled Coxeter groups, and we provide a class of reconstructible finite graphs. We show that if Γ is a finite graph which is the 1-skeleton of some simplicial flag complex L which is a homology manifold of dimension …
In this paper, we define the curvature dimension inequalities CD(m, K) on finite directed graphs modifying the case of undirected graphs. As a main result, we evaluate m and K on finite directed graphs.
We prove that there is an algorithm to determine if a given finite graph is an induced subgraph of a given curve graph.
This paper addresses questions of quasi-isometric rigidity and classification for fundamental groups of finite graphs of groups, under the assumption that the Bass-Serre tree of the graph of groups has finite depth. The main example of a finite depth graph of groups is one whose vertex and edge groups are coarse Poinca…
Study Kazdan-Warner equations on graphs using Brouwer degree theory.
problem Proving existence of solutions to Kazdan-Warner equations on finite graphs.
method Degree theory approach to uniformly bound and compute Brouwer degree.
result New proofs of existence results for Kazdan-Warner equations.
Project infinite time series graphs to finite marginal models using number theory.
problem Handling infinite time series graphs for causal inference.
method Projection method using number theory to find common ancestors in infinite graphs.
result Developed algorithm to project infinite graphs to finite marginal models.
Classifies spatial graphs with finite N-quandles.
problem Determining isomorphism of spatial graphs' N-quandles.
method Generalized N-quandles to spatial graphs, proving basic results and conjecturing a classification.
result Verifies conjecture in several cases, presents a possible counterexample.
Let Γ be a finite graph and let Γe be its extension graph. We inductively define a sequence {Γi} of finite induced subgraphs of Γe through successive applications of an operation called "doubling along a star". Then we show that every finite induced subgraph of Γe is iso…
A new layer learns abstract relations from graph structure using finite-state automata.
problem Learning abstract relations from graph structure for program analysis.
method Relaxing the problem into learning finite-state automata policies on a graph-based POMDP and training these policies using implicit differentiation.
result GFSA layer finds shortcuts in grid-world graphs and reproduces simple static analyses on Python programs.
We show that minimal length carrier graphs are not unique, but if M is in a large class of hyperbolic 3-manifolds, including the geometrically finite ones, then M has only finitely many minimal length carrier graphs and no two of them are homotopic. As a corollary, we obtain a new proof that the isometry group of a geo…
The paper proves diameter bounds and finiteness for amply regular graphs.
problem Proving diameter bounds and finiteness for amply regular graphs.
method Improved curvature estimates and new Bakry-Émery curvature estimates.
result There are only finitely many amply regular graphs with specific parameters.
Embeddings of mapping tori for end-periodic graph maps are proven.
problem Embedding mapping tori of end-periodic graph maps into finite complexes.
method Flowline-preserving homotopy equivalence and π1-injective map. result Every mapping class of Γ arising from an end-periodic homotopy equivalence contains a representative whose mapping torus realizes such an embedding.
The paper explores heat flow and constants on graphs, proving properties and proposing new concepts.
problem Analyzing heat flow and constants on graphs.
method Introducing concepts, recalling graph theory, and proposing new discrete Morse flows.
result Weak discrete Morse flows for heat flow on finite graphs under suitable assumptions.
Holomorphic analogs of Feynman integrals are shown to be finite.
problem Finite evaluation of holomorphic Feynman integrals.
method Compactification of graph moduli space with metrics.
result Holomorphic Feynman integrals are ultraviolet finite.
We embed arbitrary groups into regular graphs with prescribed automorphisms.
problem Embedding arbitrary groups into regular graphs with specific automorphisms.
method Constructing regular graphs with strong embeddings and automorphism groups isomorphic to any given finite group.
result For every d≥3 and every finite group G, there exists a d-regular graph Γ with a strong embedding β such that Aut(Γ)≅Aut(β(Γ))≅G. In this paper we study the gradient estimate for positive solutions of Schrodinger equations on locally finite graph. Then we derive Harnack's inequality for positive solutions of the Schrodinger equations. We also set up some results about Green functions of the Laplacian equation on locally finite graph. Interesting …
The curve graphs are not locally finite. In this paper, we show that the curve graphs satisfy a property which is equivalent to graphs being uniformly locally finite via Masur--Minsky's subsurface projections. As a direct application of this study, we show that there exist computable bounds for Bowditch's slices on tig…
We prove a strong form of finite rigidity for pants graphs of spheres. Specifically, for any n≥4, we construct a finite subgraph Xn of the pants graph P(S0,n) of the n-punctured sphere S0,n with the following property. Any simplicial embedding of Xn into any pants graph P(S0,m) of a punctured …
Proposes a new model for traffic flow on directed graphs.
problem Modeling advection on directed graphs for traffic flow.
method Reformulates graph advection operator as finite difference scheme; proposes DGAMGP model.
result Effective modeling of traffic flow and uncertainty as an advective process.
Graph Laplacians and machine learning predict properties of finite graphs.
problem Understanding properties of finite graphs using spectral and topological methods.
method Combining graph Laplacians, spectral inequalities, machine learning, and topological data analysis.
result Neural networks can accurately predict graph properties like Ricci-flatness and spectral gaps.
In geometric group theory one uses group actions on spaces to gain information about groups. One natural space to use is the Cayley graph of a group. The Cayley graph arguments that one encounters tend to require local finiteness, and hence finite generation of the group. In this paper, I take the theory of intersectio…
The study examines groups acting loxodromically on hyperbolic graph products.
problem Understanding groups acting loxodromically on hyperbolic graph products.
method Examined groups acting on finite products of hyperbolic graphs, focusing on loxodromic elements.
result Strong structure theorems for groups in this subclass, excluding mapping class groups of genus at least 3 and certain automorphism groups.
Study shows surfaces without certain curves have infinite orbit graph.
problem Characterizing surfaces with specific curve properties.
method Utilized tools from mapping class group geometry.
result Infinite-invariance index 1 surfaces lack good curve graphs.
In this paper, we consider three typical problems on a locally finite connected graph. The first one is to study the Bochner formula for the Laplacian operator on a locally finite connected graph. We use the Bochner formula to derive the Bernstein type estimate of the heat equation. The second is to derive the Reilly t…
Carrier graphs were first introduced for closed hyperbolic 3-manifolds by White. In this paper, we first generalize this definition to carrier graphs for representations of a rank two free group into the isometry group of hyperbolic three space. Then we prove the existence and the finiteness of minimal carrier graphs f…
Graph conditions ensure matching arc complexes are connected and hyperbolic.
problem Conditions for connectedness and hyperbolicity of matching arc complexes.
method Conditions on finite simplicial graphs guaranteeing connectedness and hyperbolicity of matching arc complexes.
result Conditions on finite simplicial graphs ensure connectedness and hyperbolicity of matching arc complexes.
An inaccessible, vertex transitive, locally finite graph is described. This graph is not quasi-isometric to a Cayley graph.
Given an edge-independent random graph G(n,p), we determine various facts about the cohomology of graph products of groups for the graph G(n,p). In particular, the random graph product of a sequence of finite groups is a rational duality group with probability tending to 1 as n goes to infinity. This includes random ri…
In this paper we study the Kato' inequality on locally finite graph. We also study the application of Kato inequality to Ginzburg-Landau equations on such graphs. Interesting properties of Schrodinger equation and a Liouville type theorem are also derived.
Every graph can be represented as a singular set of a special surface.
problem Representing any finite graph as the singular set of a compact 3D surface.
method Constructing a calibrated 3-dimensional homologically area minimizing surface with a special Lagrangian form.
result The singular set of the surface is precisely the given graph.
For given closed orientable 3-manifolds M and N let cD(M,N) be the set of mapping degrees from M to N. We address the problem: For which N, cD(M,N) is finite for all M? The answer is known in Thurston's picture of closed orientable irreducible 3-manifolds unless the target is a non-trivial graph manifol…
This is a survey on upper and lower bounds for finite group actions on bounded surfaces, 3-dimensional handlebodies and closed handles, handlebodies in arbitrary dimensions and finite graphs (the common feature of these objects is that all have free fundamental group).
The paper studies the graph geometry of finite groups, creating a dataset and analyzing its properties.
problem Understanding how group-theoretic structure is reflected in Cayley graph observables.
method Construction of a dataset of Cayley graphs for groups of order up to 767, analysis of graph statistics, and comparison of model performance.
result Graph statistics are highly informative for predicting group properties, and GNNs can recover substantial structural signal.
Automorphisms and subdivisions of Helly graphs are studied, leading to explicit models and rational translation lengths.
problem Understanding automorphisms and subdivisions of Helly graphs.
method Simple fine simplicial subdivisions and explicit simplicial models of the injective hull.
result Any automorphism of a Helly graph is either elliptic or hyperbolic, with rational translation lengths.
Research shows finiteness in triangulations with girth constraints.
problem Finiteness of cellular partial triangulations with girth constraints.
method Characterization of sparse graphs and contraction-minimal graphs.
result There are finitely many (3,6)-tight and (3,3)-tight graphs.
Researchers prove the automorphism group of a sphere complex matches the mapping group of a graph.
problem Understanding the automorphisms of sphere complexes associated with graphs.
method Analyzing the group of proper homotopy equivalences and constructing an exhaustion of the sphere complex.
result The automorphism group of the sphere complex is isomorphic to the mapping group of the graph.