Cheeger Constant Research Articles

On the optimality of J. Cheeger and P. Buser inequalities

We study the relationship between the first eigenvalue of the Laplacian and Cheeger constant when the Cheeger constant converges to zero, in the case of compact Riemannian manifolds and of finite graphs.

The Cheeger Constant, the Heat Kernel, and the Green Kernel of an Infinite Graph

This paper gives a new approach to estimate the Cheeger constant, the heat kernel, and the Green kernel of the combinatorial Laplacian for an infinite graph.

Isoperimetric properties of weighted graphs related to the Laplacian spectrum and canonical correlations

The relation between isoperimetric properties and Laplacian spectra of weighted graphs is investigated. The vertices are classified into k clusters with „few" inter-cluster edges of „small" weights (area) and „similar" cluster sizes (volumes). For k=2 the Cheeger constant represents the minimum requirement for the area/volume ratio and it is estimated from above by v?1(2-?1), where ?1 is the smallest positive eigenvalue of the weighted Laplacian. For k?2 we define the k-density of a weighted graph that is a generalization of the Cheeger constant and estimated from below by Si=1k-1?i and from above by c2 Si=1k-1 ?i, where 0<?1=…=Sk-1 are the smallest Laplacian eigenvalues and the constant c?1 depends on the metric classification properties of the corresponding eigenvectors. Laplacian spectra are also related to canonical correlations in a probabilistic setup.

Cheeger Constant and Connectivity of Graphs

In the branches of both differential geometry and graph theory, Cheeger constant plays a central role in the study of eigenvalues of Laplacians. In this paper, we give a new aspect of Cheeger constant of graphs, that is, some relations of Cheeger constant and connectivities of digraphs and graphs.

The Ising model on diluted graphs and strong amenability

Say that a graph has persistent transition if the Ising model on the graph can exhibit a phase transition (nonuniqueness of Gibbs measures) in the presence of a nonzero external field.We show that for nonamenable graphs, for Bernoulli percolation with $p$ close to 1, all the infinite clusters have persistent transition.On the other hand, we show that for transitive amenable graphs, the infinite clusters for any stationary percolation do not have persistent transition. This extends a result of Georgii for the cubic lattice. A geometric consequence of this latter fact is that the infinite clusters are strongly amenable (i.e., their anchored Cheeger constant is 0). Finally we show that the critical temperature for the Ising model with no external field on the infinite clusters of Bernoulli percolation with parameter $p$, on an arbitrary bounded degree graph, is a continuous function of $p$.

Every Graph with a Positive Cheeger Constant Contains a Tree with a Positive Cheeger Constant

It is shown that every (infinite) graph with a positive Cheeger constant contains a tree with a positive Cheeger constant. Moreover, for every nonnegative integer k there is a unique connected graph T(k) that has Cheeger constant k, but removing any edge from it reduces the Cheeger constant. This minimal graph, T(k), is a tree, and every graph G with Cheeger constant \( h(G) \geq k \) has a spanning forest in which each component is isomorphic to T(k).

Isoperimetric inequalities and the gap between the first and second eigenvalues of an Euclidean domain

In this paper we introduce a weighted Cheeger constant and show that the gap between the first two eigenvalues of a Riemannian manifold given Dirichlet conditions can be bounded from below in terms of this constant. When the Riemannian manifold is a bounded Euclidean domain satisfying an interior rolling sphere condition we give an estimate on the weighted Cheeger constant in terms of the rolling sphere radius, volume, a bound on the principal curvatures of the boundary and the dimension. This yields a lower bound on the nontrivial gap for Euclidean domains.

Some examples inL p spectral geometry

We construct surfaces $$T_{x_0 }^p $$ with volume, first eigenvalue, andL p norm of the curvature bounded independent ofx 0, but whose Cheeger constant tends to 0 asx 0 tends to infinity.

Constante de Cheeger et première valeur propre non nulle du Laplacien d’un graphe de petite dimension

Constante de Cheeger et première valeur propre non nulle du Laplacien d’un graphe de petite dimension

Graphs with parallel mean curvature

We prove that if the graph Γ f = { ( x , f ( x ) ) : x ∈ M } {\Gamma _f} = \left \{ {\left ( {x,f\left ( x \right )} \right ):x \in M} \right \} of a map f : ( M , g ) → ( N , h ) f:\left ( {M,g} \right ) \to \left ( {N,h} \right ) between Riemannian manifolds is a submanifold of ( M × N , g × h ) \left ( {M \times N,g \times h} \right ) with parallel mean curvature H H , then on a compact domain D ⊂ M D \subset M , ‖ H ‖ \left \| H \right \| is bounded from above by 1 m A ( ∂ D ) V ( D ) \frac {1}{m}\frac {{A\left ( {\partial D} \right )}}{{V\left ( D \right )}} . In particular, Γ f {\Gamma _f} is minimal provided M M is compact, or noncompact with zero Cheeger constant. Moreover, if M M is the m m -hyperbolic space—thus with nonzero Cheeger constant—then there exist real-valued functions the graphs of which are nonminimal submanifolds of M × R M \times \mathbb {R} with parallel mean curvature.

Alcohol, alcoholic...

We study the p-independence of spectra of Laplace operators on graphs arising from regular Dirichlet forms on discrete spaces. Here, a sufficient criterion is given solely by a uniform subexponential growth condition. Moreover, under a mild assumption on the measure we show a one-sided spectral inclusion without any further assumptions. We study applications to normalized Laplacians including symmetries of the spectrum and a characterization for positivity of the Cheeger constant. Furthermore, we consider Laplacians on planar tessellations for which we relate the spectral p-independence to assumptions on the curvature.