Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Metric graphs of negative type

Rutger Campbell, Kevin Hendrey, Ben Lund, Casey Tompkins

TL;DR

The paper addresses whether theta-containing metric graphs can have negative type, extending prior results that theta-free graphs are negative type; it introduces the negative type gap $\\Gamma(M,d)$ and proves a universal lower bound $\\Gamma_{\\inf}(\\mathcal{G})\\ge 1/432$ for theta-containing graphs with edge lengths at least $1$, via a specific six-point witness. It analyzes subdivisions, showing that $180$-subdivisions of graphs with a $K_{2,3}$ minor create the required theta and destroy negative type, while the $2$-subdivision of $K_4$ is $\\ell_1$-embeddable, and it conjectures that no graph with a $K_{2,3}$ minor has a $3$-subdivision whose shortest-path metric has negative type. The main theorem is established through a minimal-theta argument that excludes the existence of negative type in theta-containing metric graphs, yielding a clear dichotomy relative to theta-freeness for these spaces.

Abstract

The negative type inequalities of a metric space are closely tied to embeddability. A result by Gupta, Newman, and Rabinovich implies that if a metric graph $G$ does not contain a theta submetric as an embedding, then $G$ has negative type. We show the converse: if a metric graph $G$ contains a theta, then it does not have negative type.

Metric graphs of negative type

TL;DR

The paper addresses whether theta-containing metric graphs can have negative type, extending prior results that theta-free graphs are negative type; it introduces the negative type gap and proves a universal lower bound for theta-containing graphs with edge lengths at least , via a specific six-point witness. It analyzes subdivisions, showing that -subdivisions of graphs with a minor create the required theta and destroy negative type, while the -subdivision of is -embeddable, and it conjectures that no graph with a minor has a -subdivision whose shortest-path metric has negative type. The main theorem is established through a minimal-theta argument that excludes the existence of negative type in theta-containing metric graphs, yielding a clear dichotomy relative to theta-freeness for these spaces.

Abstract

The negative type inequalities of a metric space are closely tied to embeddability. A result by Gupta, Newman, and Rabinovich implies that if a metric graph does not contain a theta submetric as an embedding, then has negative type. We show the converse: if a metric graph contains a theta, then it does not have negative type.
Paper Structure (4 sections, 9 theorems, 17 equations, 2 figures)

This paper contains 4 sections, 9 theorems, 17 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Negative type gap for theta-containing metric graphs
  3. Subdivisions of graphs
  4. Proof of main theorem

Key Result

theorem 1

Let $(X,d)$ be a finite metric space. We label the following conditions: We have the following chain of implications: $(i)\Rightarrow(ii)\Rightarrow(iii)\Leftrightarrow(iv)\Rightarrow(v)$.

Figures (2)

  • Figure 1: A black theta with a blue shortcut, followed by four alternate thetas formed by replacing a path in the theta by the shortcut. If the black theta is minimal, then the blue shortcut is at least as long as each path removed to obtain the alternate thetas.
  • Figure 2: The blue paths show the possible shortest paths between $y_2$ and $z_2$.

Theorems & Definitions (15)

  • theorem 1: deza1997geometry
  • theorem 2: gupta2004cuts
  • theorem 3
  • corollary 1
  • corollary 2
  • proof
  • corollary 3
  • proof
  • proposition 1
  • proof
  • ...and 5 more