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Outerplanar graphs with positive Lin-Lu-Yau curvature

George Brooks, Fadekemi Osaye, Anna Schenfisch, Zhiyu Wang, Jing Yu

TL;DR

This work determines sharp structural limits for simple outerplanar graphs with positive Lin--Lu--Yau curvature: the maximum degree satisfies $\Delta(G)\le 9$ for graphs with minimum degree at least $2$, and if the graph is maximally outerplanar then $|V(G)|\le 10$. The authors develop a local-configuration analysis grounded in the Lin--Lu--Yau curvature, using both a limit-free Laplacian formulation and explicit optimal 1-Lipschitz functions to classify edge pairs that can sustain positive curvature. The results are tight, with the fan graph on 10 vertices attaining the bounds, and the proof combines combinatorial, structural, and computational (SageMath) checks for base cases. Overall, the paper advances discrete curvature theory by providing exact bounds for positively curved outerplanar graphs and introduces a precise local-configuration framework for curvature analysis via the graph Laplacian.

Abstract

In this paper, we show that all simple outerplanar graphs $G$ with minimum degree at least $2$ and positive Lin-Lu-Yau Ricci curvature on every edge have maximum degree at most $9$. Furthermore, if $G$ is maximally outerplanar, then $G$ has at most $10$ vertices. Both upper bounds are sharp.

Outerplanar graphs with positive Lin-Lu-Yau curvature

TL;DR

This work determines sharp structural limits for simple outerplanar graphs with positive Lin--Lu--Yau curvature: the maximum degree satisfies for graphs with minimum degree at least , and if the graph is maximally outerplanar then . The authors develop a local-configuration analysis grounded in the Lin--Lu--Yau curvature, using both a limit-free Laplacian formulation and explicit optimal 1-Lipschitz functions to classify edge pairs that can sustain positive curvature. The results are tight, with the fan graph on 10 vertices attaining the bounds, and the proof combines combinatorial, structural, and computational (SageMath) checks for base cases. Overall, the paper advances discrete curvature theory by providing exact bounds for positively curved outerplanar graphs and introduces a precise local-configuration framework for curvature analysis via the graph Laplacian.

Abstract

In this paper, we show that all simple outerplanar graphs with minimum degree at least and positive Lin-Lu-Yau Ricci curvature on every edge have maximum degree at most . Furthermore, if is maximally outerplanar, then has at most vertices. Both upper bounds are sharp.
Paper Structure (5 sections, 10 theorems, 20 equations, 3 figures, 2 tables)

This paper contains 5 sections, 10 theorems, 20 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Lin--Lu--Yau Ricci curvature
  3. Proof of Theorem \ref{['thm:maxdeg']}.
  4. Exact Curvature via Local Configuration
  5. Bounding the Order

Key Result

Theorem 1

Suppose $G$ is a connected simple graph embedded into a $2$-dimensional topological manifold $\Omega$ without boundary and $G$ has minimum degree at least $3$. If $G$ has positive combinatorial curvature, then it is finite and $\Omega$ is homeomorphic to either a $2$-sphere or a projective plane. Mo

Figures (3)

  • Figure 1: Fan graph on 10 vertices with curvature of each edge computed by SageMath.
  • Figure 2: $H_{xy}$ in Case 1.
  • Figure 3: $H_{xy}$ in Case 2.

Theorems & Definitions (15)

  • Theorem 1: DeVos-Mohar07
  • Theorem 2: Lu-Wang2023+
  • Theorem 3: Lu-Wang2023+
  • Theorem 4
  • Theorem 5
  • Lemma 1: Lu-Wang2023+
  • Corollary 1
  • proof : Proof of Theorem \ref{['thm:maxdeg']}
  • Lemma 2
  • proof : Proof of Lemma \ref{['lem:exterior']}
  • ...and 5 more