Graphs with nonnegative Bakry-\'Emery curvature without Quadrilateral

TL;DR

This paper classifies certain unweighted graphs with nonnegative Bakry-Émery curvature, specifically those without quadrilaterals and with girth at least five, expanding understanding of curvature conditions in graph theory.

Contribution

It extends previous classifications to include connected unweighted normalized graphs without 4-cycles under the $CD(0, abla)$ condition, with or without degree constraints.

Findings

Classifies all connected unweighted normalized $C_4$-free graphs with $CD(0, abla)$ and minimum degree at least 2.

Provides classification for non-normalized Laplacian case without degree restrictions.

Extends curvature classification to graphs with girth at least five and no quadrilaterals.

Abstract

The definition of Ricci curvature on graphs in Bakry-\'Emery's sense based on curvature dimension condition was introduced by Lin and Yau [\emph{Math. Res. Lett.}, 2010]. Hua and Lin [\emph{Comm. Anal. Geom.}, 2019] classified unweighted graphs satisfying the curvature dimension condition $CD(0,\infty)$ whose girth are at least five. In this paper, we classify all of connected unweighted normalized $C_4$-free graphs satisfying curvature dimension condition $CD(0,\infty)$ for minimum degree at least 2 and the case with non-normalized Laplacian without degree condition..