Ricci Curvature of Strong Product Graphs

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 a nice formula for the curvatures of diagonal edges, except for a special case in terms of curvatures of the factors. This gives the curvature formulas for all 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].