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Some Examples of Graphs Suggesting That the Discrete Curvature Does Sense the Smooth One

Gökçe Çakmak, Ali Deniz, Şahin Koçak, Murat Limoncu

TL;DR

The study investigates whether discrete Bakry-Émery curvature on graphs captures the smooth curvature of ambient space forms. By analyzing unweighted discrete curvature on skeletons of regular triangular tilings with unit edge lengths and deriving the corresponding ambient curvatures from well-known formulas for spherical and hyperbolic tilings, the authors observe that the signs of discrete and smooth curvatures align and both decrease along the sphere→plane→hyperbolic sequence. The results suggest that discrete curvature partly reflects ambient curvature, though exact numerical equivalence is not achieved, motivating further refinement of models and neighborhood considerations. This supports the potential relevance of discrete Bakry-Émery curvature as an indicator of smooth geometric properties in embedded graphs and informs directions for future work.

Abstract

In this note, using some regular triangular tilings of the sphere, the Euclidean plane and the hyperbolic plane, we examine the potential relationship between their discrete Bakry - Emery curvatures and the smooth curvatures of their ambient space forms.

Some Examples of Graphs Suggesting That the Discrete Curvature Does Sense the Smooth One

TL;DR

The study investigates whether discrete Bakry-Émery curvature on graphs captures the smooth curvature of ambient space forms. By analyzing unweighted discrete curvature on skeletons of regular triangular tilings with unit edge lengths and deriving the corresponding ambient curvatures from well-known formulas for spherical and hyperbolic tilings, the authors observe that the signs of discrete and smooth curvatures align and both decrease along the sphere→plane→hyperbolic sequence. The results suggest that discrete curvature partly reflects ambient curvature, though exact numerical equivalence is not achieved, motivating further refinement of models and neighborhood considerations. This supports the potential relevance of discrete Bakry-Émery curvature as an indicator of smooth geometric properties in embedded graphs and informs directions for future work.

Abstract

In this note, using some regular triangular tilings of the sphere, the Euclidean plane and the hyperbolic plane, we examine the potential relationship between their discrete Bakry - Emery curvatures and the smooth curvatures of their ambient space forms.
Paper Structure (2 sections, 4 equations, 3 figures, 1 table)

This paper contains 2 sections, 4 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Curvature of the Regular Triangular Tilings

Figures (3)

  • Figure 1: The umbrella graph $G_{n, \rho}$
  • Figure 2: The order-3, -4, and -5 regular triangular tilings of the sphere are represented in (a), (b), (c), the order-6 regular triangular tiling of the plane is represented in (d) and the order-7, -8, and -9 regular triangular tilings of the hyperbolic plane are represented in (e), (f), (g).
  • Figure 3: The 2-step neighbourhoods of the considered graphs.

Theorems & Definitions (1)

  • Definition 1