Reflective Graphs, Ollivier curvature, effective diameter, and rigidity

TL;DR

This paper establishes a discrete Bonnet Myers type theorem linking positive Ollivier curvature to effective diameter bounds in graphs, characterizes extremal cases, and introduces the concept of reflective graphs.

Contribution

It introduces the notion of reflective graphs, classifies them, and connects Ollivier curvature with graph rigidity and diameter bounds.

Findings

Diameter bounds are attained only by specific highly symmetric graphs.

Reflective graphs are classified as Cartesian products of certain well-known graphs.

The paper extends geometric concepts to discrete graph structures.

Abstract

We give a discrete Bonnet Myers type theorem for the effective diameter assuming positive Ollivier curvature. We prove that this diameter bound is attained if and only if the graph is a cocktail party graph, a Johnson graph, a halved cube, a Schl\"afli graph, a Gosset graph, or a cartesian product of the mentioned graphs with same Ollivier curvature. As a key step in the proof, we introduce the notion of reflective graphs as graphs such that for any two neighbors there exists a certain self-inverse automorphism mapping one neighbor to another. We classify these graphs as arbitrary cartesian products of the graphs mentioned before.