Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Steinerberger Curvature and Graph Distance Matrices

Wei-Chia Chen, Mao-Pei Tsui

TL;DR

This work analyzes Steinerberger's curvature on graphs defined through the distance matrix $D$ and the equation $D w = n \mathbf{1}$, focusing on how nonnegative curvature behaves under graph-building operations and the solvability of $D x = \mathbf{1}$. It provides a streamlined proof of total-curvature invariance, shows nonnegative curvature is preserved except at a small set of joining vertices under bridging, merging, and cutting, and characterizes how $D$ and its null space transform when joining graphs. It also demonstrates constructions that yield no solution to $D x = \mathbf{1}$ after combining components, and establishes spectral-geometry bounds for the Perron eigenvector on trees, linking curvature to eigenstructure. Together, these results deepen the connection between distance-based curvature, graph operations, and linear-algebraic solvability, highlighting implications for curvature theories on graphs.

Abstract

Steinerberger proposed a notion of curvature on graphs involving the graph distance matrix (J. Graph Theory, 2023). We show that nonnegative curvature is almost preserved under three graph operations. We characterize the distance matrix and its null space after adding an edge between two graphs. Let $D$ be the graph distance matrix and $\mathbf{1}$ be the all-one vector. We provide a way to construct graphs so that the linear system $Dx = \mathbf{1}$ does not have a solution.

On Steinerberger Curvature and Graph Distance Matrices

TL;DR

This work analyzes Steinerberger's curvature on graphs defined through the distance matrix and the equation , focusing on how nonnegative curvature behaves under graph-building operations and the solvability of . It provides a streamlined proof of total-curvature invariance, shows nonnegative curvature is preserved except at a small set of joining vertices under bridging, merging, and cutting, and characterizes how and its null space transform when joining graphs. It also demonstrates constructions that yield no solution to after combining components, and establishes spectral-geometry bounds for the Perron eigenvector on trees, linking curvature to eigenstructure. Together, these results deepen the connection between distance-based curvature, graph operations, and linear-algebraic solvability, highlighting implications for curvature theories on graphs.

Abstract

Steinerberger proposed a notion of curvature on graphs involving the graph distance matrix (J. Graph Theory, 2023). We show that nonnegative curvature is almost preserved under three graph operations. We characterize the distance matrix and its null space after adding an edge between two graphs. Let be the graph distance matrix and be the all-one vector. We provide a way to construct graphs so that the linear system does not have a solution.
Paper Structure (9 sections, 11 theorems, 94 equations, 3 figures)

This paper contains 9 sections, 11 theorems, 94 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Introduction.
  3. Definitions.
  4. Main Results
  5. Total Curvature and Bonnet--Myers Sharpness
  6. Bridging, Merging, and Cutting Graphs
  7. Null Space of Graph Distance Matrix
  8. Nonexistence of Curvature
  9. Perron Eigenvector of Distance Matrix

Key Result

Theorem 1

Let $G$ be a connected graph. Suppose there are $w_1,w_2 \in {\mathbb R}^n_{\geq 0}$ so that $Dw_1=Dw_2=n\cdot\mathbf{1}.$ Then $\|w_1\|_1=\|w_2\|_1.$

Figures (3)

  • Figure 1: Adding an edge between the complete graph $K_4$ and the cycle $C_6$.
  • Figure 2: Adding an edge between two copies of $K_5$.
  • Figure 3: Merging the complete graph $K_4$ and the cycle $C_6$ at a vertex.

Theorems & Definitions (20)

  • Theorem 1: curvatureongraphs
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Theorem 2: Bridging Graphs
  • proof : Proof of Theorem 2
  • Corollary 1
  • Theorem 3: Merging Graphs
  • ...and 10 more