[Paper Review] Cheeger estimates of Dirichlet-to-Neumann operators on infinite subgraphs of graphs

Bobo Hua, Yan Huang+1 more|arXiv (Cornell University)|Oct 25, 2018

Spectral Theory in Mathematical Physics26 references4 citations

TL;DR

This paper establishes Cheeger-type estimates for the Dirichlet-to-Neumann (DtN) operator on infinite subgraphs of graphs, proving lower and upper bounds for the k-th eigenvalue in terms of higher-order Cheeger-Steklov constants. Using exhaustion methods and spectral convergence, it extends finite-graph results to infinite settings, showing that the bottom spectrum and higher eigenvalues are controlled by geometric isoperimetric quantities, with explicit universal constants in the lower bound.

ABSTRACT

In this paper, we study the Dirichlet-to-Neumann operators on infinite subgraphs of graphs. For an infinite graph, we prove Cheeger-type estimates for the bottom spectrum of the Dirichlet-to-Neumann operator, and the higher order Cheeger estimates for higher order eigenvalues of the Dirichlet-to-Neumann operator.

Motivation & Objective

• To extend Cheeger-type eigenvalue estimates for the Dirichlet-to-Neumann operator from finite to infinite subgraphs of graphs.
• To establish higher-order Cheeger estimates for the k-th eigenvalue of the DtN operator on infinite graphs using exhaustion techniques.
• To prove that the bottom spectrum and higher eigenvalues of the DtN operator are bounded below and above by geometric isoperimetric constants, generalizing finite-graph results.
• To demonstrate spectral convergence of eigenvalues on finite exhaustion sets to the infinite graph limit, ensuring stability of estimates.

Proposed method

• Constructing the Dirichlet-to-Neumann operator on infinite subgraphs via exhaustion of the graph by finite vertex sets.
• Using harmonic extensions and boundary Dirichlet-to-Neumann maps to define eigenvalue problems on finite subgraphs.
• Applying the spectral convergence of Laplacian eigenvalues on exhaustion sets to show convergence of DtN eigenvalues to the infinite graph limit.
• Employing the higher-order Cheeger inequality from Lee, Oveis Gharan, and Trevisan (2014) to bound the k-th eigenvalue in terms of k-way partitioning constants.
• Defining higher-order Cheeger-Steklov constants $ h_k^k(W) $ and $ h_k( heta) $ to quantify isoperimetric properties of subgraphs.
• Proving that the limit of eigenvalues on exhaustion sets equals the eigenvalues on the infinite graph, enabling the derivation of global estimates.

Experimental results

Research questions

• RQ1 Can Cheeger-type estimates for the Dirichlet-to-Neumann operator be extended from finite to infinite subgraphs of graphs?
• RQ2 How do the higher-order eigenvalues of the DtN operator on infinite graphs relate to geometric isoperimetric constants?
• RQ3 What is the role of exhaustion sequences in approximating the spectrum of the DtN operator on infinite graphs?
• RQ4 Can spectral convergence of finite exhaustion sets be used to derive global lower and upper bounds for eigenvalues on infinite graphs?
• RQ5 What is the dependence of the k-th eigenvalue on the k-th order Cheeger-Steklov constant in the infinite graph setting?

Key findings

• The first nontrivial eigenvalue of the DtN operator on an infinite subgraph satisfies $ \sigma_2(W) \geq \frac{1}{4} h_N(W) h_J(W) $, extending Jammes' estimate to infinite graphs.
• For the k-th eigenvalue, the paper proves $ \sigma_k(W) \geq \frac{c}{k^6} h_k(W) $, where $ h_k(W) $ is the k-th order Cheeger-Steklov constant and $ c $ is a universal constant.
• The upper bound satisfies $ \sigma_k(W) \leq h_k^k(W) $, showing tightness of the estimate in terms of the higher-order isoperimetric constant.
• Spectral convergence is established: $ \lim_{r \to \infty} \lambda^{(r)}_{k,D}(W) = \sigma_k(W) $, ensuring stability of eigenvalue estimates under exhaustion.
• The bottom spectrum of the DtN operator on an infinite graph $ \Omega $ satisfies $ \sigma_k(\Omega) \geq \frac{c}{k^6} h_k(\Omega) $ and $ \sigma_k(\Omega) \leq h_k^k(\Omega) $, with limits taken over exhaustion sequences.
• The results generalize finite-graph estimates from [HHW17] and [HM17] to the infinite setting, establishing a robust link between spectral and geometric properties.