Node resistance curvature in Cartesian graph products

TL;DR

This paper explores how node resistance curvature behaves under Cartesian graph products, revealing that interior vertices of grids tend to have nonpositive curvature while boundary vertices tend to have nonnegative curvature.

Contribution

It provides the first analysis of node resistance curvature in Cartesian products, establishing positivity and negativity conditions for grid graphs and offering bounds and open questions.

Findings

Interior vertices of m×n grids have nonpositive curvature for m,n≥3.

Boundary vertices of grids have nonnegative curvature.

Counterexamples show limitations of generalizations.

Abstract

Devriendt and Lambiotte recently introduced the \emph{node resistance curvature}, a notion of graph curvature based on the effective resistance matrix. In this paper, we begin the study of the behavior of the node resistance curvature under the operation of the Cartesian graph product. We study the natural question of global positivity of node resistance curvature of the Cartesian product of positively-curved graphs, and prove that, whenever $m,n\ge3$, the node resistance curvature of the interior vertices of a $m\times n$ grid is always nonpositive, while it is always nonnegative on the boundary of such grids. For completeness, we also prove a number of results on node resistance curvature in $2\times n$ grids and exhibit a counterexample to a generalization. We also give generic bounds and suggest several further questions for future study.