Highly regular vertex-transitive graphs are globally rigid

TL;DR

This paper proves that highly regular vertex-transitive graphs are globally rigid in any dimension, confirming a conjecture and establishing the optimal regularity constant through examples.

Contribution

It establishes that highly regular vertex-transitive graphs are globally rigid in all dimensions and provides examples demonstrating the optimality of the regularity constant.

Findings

Highly regular vertex-transitive graphs are globally rigid in any dimension.

The regularity constant for global rigidity is proven to be optimal.

Constructed examples show the bounds of the regularity condition.

Abstract

A graph is said to be globally rigid in $d$-dimensional space if almost all of its embeddings are unique up to isometries. If a graph has enough automorphisms to send any of its vertices into any other, then it is called vertex-transitive. We show that, in any dimension, highly regular vertex-transitive graphs are globally rigid, positively answering a conjecture of Sean Dewar. Furthermore, we construct examples that show that our constant for regularity is best possible.