Curvatures, graph products and Ricci flatness

TL;DR

This paper compares different Ricci curvature notions on graphs, explores their relationships with Ricci flatness, and studies how these properties behave under various graph products, providing new insights into graph curvature theory.

Contribution

It establishes conditions linking Ollivier Ricci and Bakry-Émery curvatures, analyzes their preservation under graph products, and characterizes curvature in distance-regular graphs.

Findings

Non-negativity of Ollivier Ricci implies Bakry-Émery non-negativity under certain conditions.

Ricci flatness is preserved under all natural graph products.

Distance-regular graphs of girth 4 attain maximal curvature values.

Abstract

In this paper, we compare Ollivier Ricci curvature and Bakry-\'Emery curvature notions on combinatorial graphs and discuss connections to various types of Ricci flatness. We show that non-negativity of Ollivier Ricci curvature implies non-negativity of Bakry-\'Emery curvature under triangle-freeness and an additional in-degree condition. We also provide examples that both conditions of this result are necessary. We investigate relations to graph products and show that Ricci flatness is preserved under all natural products. While non-negativity of both curvatures are preserved under Cartesian products, we show that in the case of strong products, non-negativity of Ollivier Ricci curvature is only preserved for horizontal and vertical edges. We also prove that all distance-regular graphs of girth $4$ attain their maximal possible curvature values.