The paper compares Steklov and Laplacian eigenvalues on graphs.
problem Understanding the relationship between Steklov and Laplacian eigenvalues on graphs.
method Analyzing eigenvalues and discussing rigidity.
result Obtained Lichnerowicz-type estimates and combinatorial estimates for Steklov eigenvalues.
Paper finds a graph Steklov eigenvalue estimate with rigidity results.
problem Estimating Steklov eigenvalues on graphs.
method Lichnerowicz-type estimate for the first Steklov eigenvalues.
result Rigidity results for the Steklov eigenvalues on graphs.
Paper extends Steklov eigenvalue estimate to weighted graphs.
problem Steklov eigenvalue estimation on weighted graphs.
method Extended Perrin's estimate to general weighted graphs.
result Characterized rigidity of the extended estimate.
Paper finds how Steklov eigenvalues change on graphs and trees.
problem Understanding how Steklov eigenvalues vary on graphs and trees.
method Analyzes monotonicity of Steklov eigenvalues on graphs and trees.
result Extends Steklov eigenvalue results to higher eigenvalues and trees.
Paper describes eigenvalues of genus 3 surfaces graphs.
problem Understanding eigenvalues of genus 3 surfaces.
method Analyzes graphs derived from pair of pants decompositions.
result Complete description of eigenvalue sets for genus 3.
The paper bounds higher Steklov eigenvalues of graphs on surfaces.
problem Bounding higher Steklov eigenvalues of graphs on surfaces.
method Using metrical deformation via probability flows, the upper bound is derived.
result The upper bound of higher Steklov eigenvalues is established.
The paper compares eigenvalues of Dirichlet, Neumann, and Laplacian on graphs.
problem Eigenvalue comparisons on graphs.
method Analytical comparisons and discussions of eigenvalues and their applications.
result Extensions of eigenvalue estimates for Dirichlet and Neumann eigenvalues.
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 (ε,ρ)-proximity graphs. result Eigenvalues of truncated graph Laplacians converge to Dirichlet eigenvalues of the Laplace-Beltrami operator.
The paper finds minimum Steklov eigenvalues on combinatorial graphs.
problem Finding the minimum Steklov eigenvalues on combinatorial graphs.
method Extending Friedman's nodal domain theory for Laplacian eigenfunctions to Steklov eigenfunctions.
result The minimum of the imth Steklov eigenvalue on a connected combinatorial graph is essentially attained by a star or a regular comb with minimal brooms. Edge augmentation connects disconnected graphs by elevating eigenvalues.
problem Connecting disconnected subgraphs in graphs with zero eigenvalues.
method Elevating zero eigenvalues of graph's spectrum to connect subgraphs.
result The algorithm consistently connects graph components, achieving >50% inter-community edges.
Estimates eigenvalues and spectrum for graph substructures using isocapacitary constants.
problem Estimating eigenvalues and spectrum for graph substructures.
method Introducing Cheeger type constants via isocapacitary constants to estimate eigenvalues and spectrum.
result Estimates for first Dirichlet, Neumann, and Steklov eigenvalues, as well as the bottom of the spectrum of the Laplace operator and Dirichlet-to-Neumann operator.
Graph Laplacian approximates manifold eigenvalues with controlled curvature bounds.
problem Approximating eigenvalues of Laplace-Beltrami on manifolds with bounded Ricci curvature.
method Graph discretization of Riemannian manifolds with (ε,ρ)-approximation, proving eigenvalue convergence. result Graph Laplacian eigenvalues converge uniformly to manifold Laplacian eigenvalues as parameters approach zero.
We derive the limiting distribution for the largest eigenvalues of the adjacency matrix for a stochastic blockmodel graph when the number of vertices tends to infinity. We show that, in the limit, these eigenvalues are jointly multivariate normal with bounded covariances. Our result extends the classic result of Füredi…
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.
Estimates graph curvature and diameter using Laplacian eigenvalues.
problem Estimating graph curvature and diameter using Laplacian eigenvalues.
method Combination of gradient estimates and strong nodal domain walks.
result Li-Yau type eigenvalue-diameter estimate for signed graphs.
Optimizing quantum graphs yields geodesic nets on surfaces.
problem Finding optimal quantum graphs for geodesic nets.
method Optimizing functionals from spectral theory to find geodesic nets.
result Critical metrics for eigenvalues give rise to geodesic nets.
The paper proves inequalities for Steklov eigenvalues on finite graphs.
problem Eigenvalues of Laplacians for reversible Markov chains and Steklov eigenvalues.
method Generalized Cheeger inequalities, convergence results, and resolvent convergence.
result Sharp estimate for the first non-trivial Steklov eigenvalue.
We discuss optimal lower bounds for eigenvalues of Laplacians on weighted graphs. These bounds are formulated in terms of the geometry and, more specifically, the inradius of subsets of the graph. In particular, we study the first non-zero eigenvalue in the finite volume case and the first eigenvalue of the Dirichlet L…
Study Steklov eigenvalues on hyperbolic triangle-tiling graphs.
problem Analyzing Steklov eigenvalues on specific hyperbolic graph structures.
method Introduced a graph roughly isometric to hyperbolic plane, used discretization to transfer bounds.
result Steklov eigenvalues tend to zero proportionally to the inverse of the domain size.
We show that eigenvalues and eigenfunctions of the Laplace-Beltrami operator on a Riemannian manifold are approximated by eigenvalues and eigenvectors of a (suitably weighted) graph Laplace operator of a proximity graph on an epsilon-net.
The paper studies eigenvalues of graph Laplacians on data clouds and proves central limit theorems.
problem Asymptotic fluctuations of eigenvalues of graph Laplacians on data clouds.
method Analysis of graph Laplacian operator, asymptotic fluctuations, central limit theorems.
result Central limit theorems for eigenvalues of graph Laplacians are proven.
Upper bounds for Steklov eigenvalues in subgraphs of polynomial growth Cayley graphs.
problem Finding upper bounds for Steklov eigenvalues in subgraphs of polynomial growth Cayley graphs.
method Discretizing a bounded domain and using comparison theorems.
result The $k^{\mbox{th}}$ eigenvalue tends to 0 proportionally to 1/∣B∣d−11. In this paper, we study eigenvalues and eigenfunctions of p-Laplacians with Dirichlet boundary condition on graphs. We characterize the first eigenfunction (and the maximum eigenfunction for a bipartite graph) via the sign condition. By the uniqueness of the first eigenfunction of p-Laplacian, as p→1, we ident…
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.
New curvature measure for graphs improves diameter and eigenvalue estimates.
problem Estimating properties of graphs using Ricci curvature.
method Introduced integral Ricci curvature Iκ0 for graphs. result Uniform estimates for diameter, number of vertices, and eigenvalue.
Discrete analogues of classical spectral geometric inequalities and extremal eigenvalue problems on graphs.
problem Extremal eigenvalue problems on graphs
method Developing nodal domain methods for adjacency matrices
result Establishing sharp extremal characterizations across diverse graph classes
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.
By introducing a weight function to the Laplace operator, Bakry and Émery defined the "drift Laplacian" to study diffusion processes. Our first main result is that, given a Bakry-Émery manifold, there is a naturally associated family of graphs whose eigenvalues converge to the eigenvalues of the drift Laplacian as the …
Universal inequalities for Laplacian eigenvalues on discrete groups.
problem Proving inequalities for Laplacian eigenvalues on discrete groups.
method Analyzing Laplacian eigenvalues with Dirichlet boundary conditions on subsets of discrete groups.
result Yang-type universal inequalities for Cayley graphs of amenable groups and the d-regular tree.
Graph Neural Networks outperform the Weisfeiler-Lehman algorithm in representation power.
problem Limited representation power of Graph Neural Networks compared to the Weisfeiler-Lehman algorithm.
method Algebraic analysis using eigenvalue decomposition of graph operators.
result Graph Neural Networks produce more discriminative representations than the Weisfeiler-Lehman algorithm.
Study of endperiodic maps on infinite graphs, proving homotopy and eigenvalue properties.
problem Understanding endperiodic maps on infinite graphs with finitely many ends.
method Adapting relative train track maps and combinatorial techniques to infinite type setting.
result Any generalized endperiodic map is homotopic to a relative train track map.
Using expander graphs, we construct a sequence of smooth compact surfaces with boundary of perimeter N, and with the first non-zero Steklov eigenvalue uniformly bounded away from zero. This answers a question which was raised in [9]. The genus grows linearly with N, this is the optimal growth rate.
A novel hypergraph partitioning method using tensor eigenvalue decomposition captures super-dyadic interactions.
problem Capturing super-dyadic interactions in k-uniform hypergraphs.
method Tensor-based representation and tensor eigenvalue decomposition for capturing interactions.
result Improved min-cut solution on 2-uniform hypergraphs (graphs) compared to standard spectral partitioning.
How does coarsening affect the spectrum of a general graph? We provide conditions such that the principal eigenvalues and eigenspaces of a coarsened and original graph Laplacian matrices are close. The achieved approximation is shown to depend on standard graph-theoretic properties, such as the degree and eigenvalue di…
We define the distance between edges of graphs and study the coarse Ricci curvature on edges. We consider the Laplacian on edges based on the Jost-Horak's definition of the Laplacian on simplicial complexes. As one of our main results, we obtain an estimate of the first non-zero eigenvalue of the Laplacian by the Ricci…
Study on harmonic maps between cones, linking degrees to graph Laplacian eigenvalues.
problem Understanding harmonic maps between singular spaces.
method Analyzing homogeneous harmonic maps between simplicial cones and their degrees.
result Degrees of homogeneous harmonic maps are related to eigenvalues of discrete graph Laplacians.
Proposes a faster Isomap algorithm by reducing eigenvalue decomposition complexity.
problem High computational complexity of Isomap, especially in eigenvalue decomposition stage.
method Introduces a projection operator to reduce the complexity of the eigenvalue decomposition stage to linear order.
result Reduces Isomap's computational complexity to linear order while preserving structural information.
The paper proves spectral convergence rates for graph Laplacian to manifold Laplace-Beltrami operator.
problem Spectral convergence of graph Laplacian to manifold Laplace-Beltrami operator.
method Analysis of Dirichlet form convergence and construction of approximate eigenfunctions via manifold heat kernel.
result Proves spectral convergence rates for Gaussian kernelized graph Laplacian.
In this paper, we first introduce higher order Dirichlet-to-Neumann maps on graphs which can be viewed as a discrete analogue of the corresponding Dirichlet-to-Neumann maps on compact Riemannian manifolds with boundary and a higher order generalization of the Dirichlet-to-Neumann map on graphs introduced by Hua-Huang-W…
The paper defines surface area for graphs and derives spectral estimates.
problem Understanding connectivity measures and spectral properties of graphs.
method Introducing surface area concepts related to inverse degree and deriving spectral bounds.
result An upper bound on the second eigenvalue for planar graphs.
We compute an approximate Fréchet mean for sets of sparse graphs.
problem Characterizing the location of a set of graphs in a metric space.
method We use the pseudometric defined by the ℓ₂ norm of eigenvalues of adjacency matrices.
result We describe an algorithm to approximate the Fréchet mean of a set of graphs.
The spectral geometry of mesh matrices of graphs is explored, leading to new formulas and eigenvalue estimates.
problem Understanding the spectral properties of mesh matrices of graphs.
method Definition and study of mesh matrices, introduction of mesh Laplacian, derivation of characteristic polynomial formulas.
result Mesh Laplacian eigenvalues are all real and greater than or equal to 1, with a smallest positive eigenvalue estimated.
The study shows how discrete graphs can resemble hypercube structures under certain curvature conditions.
problem Understanding the structure of graphs with specific curvature conditions.
method Analyzing weighted graphs with lower Ricci curvature bounds and eigenvalue closeness to establish structural similarity.
result Discrete graphs with specific curvature conditions are close to hypercube structures in terms of Frobenius distance and eigenfunctions.
Estimates eigenvalues of poly-Laplace operator on lattice subgraphs.
problem Estimating eigenvalues of poly-Laplace operator on subgraphs of lattice graphs.
method Introduced discrete poly-Laplace operator, derived upper and lower bounds for eigenvalues.
result Poly-Laplace eigenvalues are at least squares of lower-order poly-Laplace eigenvalues.
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.
Suppose that G=(V,E) is a connected locally finite graph with the vertex set V and the edge set E. Let Ω⊂V be a bounded domain. Consider the following quasilinear elliptic equation on graph G $$ \left \{ \begin{array}{lcr} -Δ_{p}u= λK(x)|u|^{p-2}u+f(x,u), \ \ x\inΩ^{\circ}, u=0, \ \ x\in\partial Ω, \\…
We present a method for proving upper bounds on the eigenvalues of the graph Laplacian. A main step involves choosing an appropriate "Riemannian" metric to uniformize the geometry of the graph. In many interesting cases, the existence of such a metric is shown by examining the combinatorics of special types of flows. T…
We study the convergence of the graph Laplacian of a random geometric graph generated by an i.i.d. sample from a m-dimensional submanifold M in Rd as the sample size n increases and the neighborhood size h tends to zero. We show that eigenvalues and eigenvectors of the graph Laplacian converge with a rate of…