Ollivier-Ricci curvature of regular graphs

TL;DR

This paper derives explicit formulas for Ollivier-Ricci and Lin-Lu-Yau curvature in regular graphs, enabling efficient computation, characterization of special edges, and establishing conditions for positive curvature and graph classification.

Contribution

It provides explicit formulas for curvatures in regular graphs, characterizes bone idle edges, and proves a rigidity theorem for cocktail party graphs.

Findings

Explicit formulas for curvatures in regular graphs.

Characterization of bone idle edges and graphs.

Rigidity theorem for cocktail party graphs.

Abstract

We derive explicit formulas for the Lin-Lu-Yau curvature and the Ollivier-Ricci curvature in terms of graph parameters and an optimal assignment. Utilizing these precise expressions, we examine the relationship between the Lin-Lu-Yau curvature and the 0-Ollivier-Ricci curvature, resulting in an equality condition on regular graphs. This condition allows us to characterize edges that are bone idle in regular graphs of girth four and to construct a family of bone idle graphs with this girth. We then use our formulas to provide an efficient implementation of the Ollivier-Ricci curvature on regular graphs, enabling us to identify all bone idle, regular graphs with fewer than 15 vertices. Moreover, we establish a rigidity theorem for cocktail party graphs, proving that a regular graph is a cocktail party graph if and only if its Lin-Lu-Yau curvature is equal to one. Furthermore, we present a condition on the degree of a regular graph that guarantees positive Ricci curvature. We conclude this work by discussing the maximal number of vertices that a regular graph of fixed degree with positive Lin-Lu-Yau curvature can have.