Ollivier curvature, betweenness centrality and average distance

TL;DR

This paper establishes a new upper bound for average graph distance using Ollivier curvature weighted by edge betweenness centrality, with equality characterized by specific highly symmetric graphs.

Contribution

It introduces a novel upper bound relating average graph distance to Ollivier curvature weighted by betweenness centrality and characterizes cases of equality.

Findings

Derived a new upper bound for average graph distance.

Identified conditions for equality involving specific graph classes.

Connected curvature, centrality, and distance in graph theory.

Abstract

We give a new upper bound for the average graph distance in terms of the average Ollivier curvature. Here, the average Ollivier curvature is weighted with the edge betweenness centrality. Moreover, we prove that equality is attained precisely for the reflective graphs which have been classified as Cartesian products of cocktail party graphs, Johnson graphs, halved cubes, Schl\"afli graphs, and Gosset graphs.