[Paper Review] Cheeger constants, growth and spectrum of locally tessellating planar graphs

Matthias Keller, Norbert Peyerimhoff|ArXiv.org|Mar 27, 2009

Advanced Graph Theory Research13 references4 citations

TL;DR

This paper establishes explicit lower bounds for the Cheeger constants of locally tessellating planar graphs using combinatorial curvature, linking local geometric properties to global invariants like exponential growth and spectral gaps. It proves that strictly positive vertex curvature or normalized curvature implies positive Cheeger constants, with sharp estimates depending on vertex and face degree bounds, and uses a tree-comparison construction to relate growth rates to spectral properties.

ABSTRACT

In this article, we study relations between the local geometry of planar graphs (combinatorial curvature) and em global geometric invariants, namely the Cheeger constants and the exponential growth. We also discuss spectral applications.

Motivation & Objective

• To relate local combinatorial curvature to global geometric invariants such as Cheeger constants and exponential growth in planar graphs.
• To derive explicit lower bounds for the physical and geometric Cheeger constants in terms of curvature and degree constraints.
• To prove that strictly positive curvature implies positive Cheeger constants, generalizing discrete Bonnet-Myers and Cartan-Hadamard theorems.
• To show that exponential growth in such graphs is bounded below by that of a comparison tree, using a face-opening construction.
• To connect curvature conditions to spectral properties via Cheeger constants, particularly for Laplacian spectra.

Proposed method

• Define combinatorial curvature at vertices and corners using face and vertex degrees, with infinigons treated as having infinite degree.
• Introduce two Cheeger constants: α(G) for physical growth (edge boundary over vertex count), and ḡα(G) for geometric growth (edge boundary over volume).
• Establish curvature-based lower bounds via the constant $ C_{p,q} $, which depends on vertex and face degree bounds $ p $ and $ q $.
• Use a constructive procedure to transform finite faces into infinigons by cutting along geodesic rays, thereby creating a comparison tree T with bounded vertex degree.
• Prove that the exponential growth rate of the original graph is bounded above by that of the comparison tree, preserving or increasing growth rate.
• Apply the tree-comparison method to derive lower bounds on Cheeger constants and relate them to spectral gaps via known Cheeger inequalities.

Experimental results

Research questions

• RQ1 How do local curvature conditions on planar graphs constrain their global Cheeger constants?
• RQ2 What is the relationship between combinatorial curvature and exponential growth in locally tessellating planar graphs?
• RQ3 Can the Cheeger constant be bounded from below using only vertex and face degree bounds and curvature infima?
• RQ4 How does the structure of finite faces affect the growth and spectral properties of planar graphs?
• RQ5 To what extent can a comparison tree with the same degree bounds capture the growth and spectral behavior of a general planar graph?

Key findings

• The physical Cheeger constant satisfies $ \alpha(G) \geq 2C_{p,q}C $, where $ C = \inf_v (-\kappa(v)) > 0 $, with $ C_{p,q} $ defined by degree constraints.
• The geometric Cheeger constant satisfies $ \widetilde{\alpha}(G) \geq 2C_{p,q}c $, where $ c = \inf_v \left(-\frac{1}{|v|}\kappa(v)\right) > 0 $, under the same conditions.
• The bounds are sharp in the case of regular trees, where $ q = \infty $, confirming optimality in extremal cases.
• Exponential growth $ \mu(G) $ is bounded above by the growth of a $ p $-regular tree, with equality if curvature is non-negative and the graph is a tree.
• The construction of a comparison tree by opening finite faces into infinigons preserves or increases growth rate, proving $ \mu(G) \leq \mu(T) $ for a $ p $-bounded tree $ T $.
• The results generalize discrete Bonnet-Myers and Cartan-Hadamard theorems, showing that positive curvature implies finite graphs and non-positive curvature implies no cut locus.