Graphs with Lin-Lu-Yau curvature at least one and regular bone-idle graphs

TL;DR

This paper characterizes graphs with Lin-Lu-Yau curvature at least one, explores their relationship with Ollivier-Ricci curvature, and fully classifies regular bone-idle graphs, including non-existence results for certain degrees.

Contribution

It provides a complete characterization of graphs with high Lin-Lu-Yau curvature and classifies all regular bone-idle graphs, including non-existence results for specific degrees.

Findings

Graphs with Lin-Lu-Yau curvature ≥ 1 are fully characterized.

Exact formulas relate Lin-Lu-Yau and Ollivier-Ricci curvatures on regular graphs.

3-regular bone-idle graphs do not exist; all 4-regular bone-idle graphs are characterized.

Abstract

We study the Ollivier-Ricci curvature and its modification introduced by Lin, Lu, and Yau on graphs. We provide a complete characterization of all graphs with Lin-Lu-Yau curvature at least one. We then explore the relationship between the Lin-Lu-Yau curvature and the Ollivier-Ricci curvature with vanishing idleness on regular graphs. An exact formula for the difference between these two curvature notions is established, along with an equality condition. This condition allows us to characterize edges that are bone-idle in regular graphs. Furthermore, we demonstrate the non-existence of 3-regular bone-idle graphs and present a complete characterization of all 4-regular bone-idle graphs. We also show that there exist no 5-regular bone-idle graphs that are symmetric or a Cartesian product of a 3-regular and a 2-regular graph.