Reflective Graphs, Ollivier curvature, effective diameter, and rigidity
Combinatorics
2022-06-01 v1 Differential Geometry
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.
Keywords
Cite
@article{arxiv.2205.15857,
title = {Reflective Graphs, Ollivier curvature, effective diameter, and rigidity},
author = {Florentin Münch},
journal= {arXiv preprint arXiv:2205.15857},
year = {2022}
}