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Classifying the globally rigid edge-transitive graphs and distance-regular graphs in the plane

Sean Dewar

Abstract

A graph is said to be globally rigid if almost all embeddings of the graph's vertices in the Euclidean plane will define a system of edge-length equations with a unique (up to isometry) solution. In 2007, Jackson, Servatius and Servatius characterised exactly which vertex-transitive graphs are globally rigid solely by their degree and maximal clique number, two easily computable parameters for vertex-transitive graphs. In this short note we will extend this characterisation to all graphs that are determined by their automorphism group. We do this by characterising exactly which edge-transitive graphs and distance-regular graphs are globally rigid by their minimal and maximal degrees.

Classifying the globally rigid edge-transitive graphs and distance-regular graphs in the plane

Abstract

A graph is said to be globally rigid if almost all embeddings of the graph's vertices in the Euclidean plane will define a system of edge-length equations with a unique (up to isometry) solution. In 2007, Jackson, Servatius and Servatius characterised exactly which vertex-transitive graphs are globally rigid solely by their degree and maximal clique number, two easily computable parameters for vertex-transitive graphs. In this short note we will extend this characterisation to all graphs that are determined by their automorphism group. We do this by characterising exactly which edge-transitive graphs and distance-regular graphs are globally rigid by their minimal and maximal degrees.
Paper Structure (7 sections, 19 theorems, 5 equations, 3 figures)

This paper contains 7 sections, 19 theorems, 5 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Combinatorial rigidity and global rigidity
  3. Edge-transitive graphs
  4. Distance-regular graphs
  5. Classifying the rigid but not globally rigid graphs
  6. A conjecture for higher dimensional global rigidity
  7. The graph H(6,10)

Key Result

Theorem 1

Let $G=(V,E)$ be a connected vertex-transitive graph with degree $k$. Then $G$ is globally rigid if and only if either:

Figures (3)

  • Figure 1: The three special cases of \ref{['t:maine']}. The top two graphs are the complete bipartite graphs $K_{3,4}$ and $K_{3,5}$ respectively. The bottom graph, which we call $H_{6,10}$, is described fully in \ref{['appendix']}.
  • Figure 2: A 7-regular not globally rigid graph formed from two copies of $K_8$ by deleting an edge from each and joining them together
  • Figure 3: The only two cubic graphs that are rigid but not complete; $K_{3,3}$ (left) and $K_2 \times K_3$ (right).

Theorems & Definitions (32)

  • Theorem 1: JSS07
  • Theorem 2
  • Theorem 3
  • Theorem 4: Poll1927
  • Lemma 5: see jordan
  • Theorem 6: JaJo05
  • Theorem 7: LoYe82
  • Theorem 8: JSS07
  • Theorem 9: JSS07
  • Theorem 10: Watkins
  • ...and 22 more