Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Node resistance curvature in Cartesian graph products

Aleyah Dawkins, Vishal Gupta, Mark Kempton, William Linz, Jeremy Quail, Harry Richman, Zachary Stier

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.

Node resistance curvature in Cartesian graph products

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 , the node resistance curvature of the interior vertices of a 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 grids and exhibit a counterexample to a generalization. We also give generic bounds and suggest several further questions for future study.
Paper Structure (9 sections, 9 theorems, 16 equations, 6 figures)

This paper contains 9 sections, 9 theorems, 16 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Background and notation
  3. Cartesian graph products
  4. Basic properties of electrical resistance
  5. Products of paths
  6. Wide grids and wide path products
  7. Ladder graphs
  8. General bounds and future questions
  9. Code

Key Result

Theorem 6

Consider the grid graph $P_m\square P_n$ with $m,n\ge3$. Then, the interior vertices all have negative node resistance curvature, and the boundary vertices all have nonnegative node resistance curvature. If further either $m>3$ or $n>3$ then the node resistance curvature on the boundary is positive

Figures (6)

  • Figure 1: The $3\times3$ box (orange) is slid until it contains the target vertex (red).
  • Figure 2: Boundary curvatures for the $3\times3$ grid. Produced with the Mathematica command GridGraph[3, 3, VertexLabels -> GridBoundaryNodeCurvatures[3, 3]].
  • Figure 3: Boundary curvatures for the $3\times4$ grid. Produced with the Mathematica command GridGraph[3, 4, VertexLabels -> GridBoundaryNodeCurvatures[3, 4]].
  • Figure 4: Examples of rungs and rails in ladders, highlighted in red.
  • Figure 5: Possibilities for edges incident to vertices in ladders, highlighted in red.
  • ...and 1 more figures

Theorems & Definitions (18)

  • Example 2
  • Example 3
  • Conjecture 4
  • Conjecture 5
  • Theorem 6
  • Proposition 7: Rayleigh's monotonicity law, cf. DS
  • Proposition 8: Curvature monotonicity
  • Proposition 9
  • proof : Proof of \ref{['thm:grid curv']}
  • Proposition 10
  • ...and 8 more