Ricci Curvature of Strong Product Graphs

TL;DR

This paper derives explicit Ricci curvature formulas for the strong product of regular graphs, clarifies the case of diagonal edges, and simplifies existing formulas for Cartesian products, advancing understanding of graph curvature.

Contribution

It provides the first explicit curvature formulas for strong product graphs and simplifies existing results for Cartesian products.

Findings

Explicit curvature formulas for horizontal and vertical edges in strong product graphs.

No general formula exists for diagonal edges, except in special cases.

A sharper lower bound for diagonal edge curvatures in terms of factor curvatures.

Abstract

We establish for the first time the explicit curvature formulas for the horizontal and vertical edges of the strong product of two regular graphs. We complement this result with showing that there does not exist an analogous formula for the curvatures of diagonal edges except for a special case, and providing a sharp lower bound for them in terms of the curvatures of the factors. This gives the curvature formulas for all the edges of the product of a complete graph and a regular graph. We also present an accessible and simpler proof of the curvature formulas for all the edges of the Cartesian product of two regular graphs, originally established by Lin, Lu, and Yau [2011].