Flexibility and movability in Cayley graphs

TL;DR

This paper investigates the properties of flexibility and movability in Cayley graphs, providing constructions of infinite families of movable graphs and explicit examples of dense, movable graphs.

Contribution

It introduces the concepts of flexibility and movability in graphs, specifically focusing on Cayley graphs, and constructs infinite and dense examples of movable graphs.

Findings

Constructed regular moving graphs of all degrees

Provided explicit dense, movable graph constructions

Analyzed conditions for flexibility and movability in Cayley graphs

Abstract

Let $\mathbf{\Gamma} = (V,E)$ be a (non-trivial) finite graph with $\lambda: E \rightarrow \mathbb{R}_{+}$, an edge labelling of $\mathbf{\Gamma}$. Let $\rho : V\rightarrow \mathbb{R}^{2}$ be a map which preserves the edge labelling. The graph $\mathbf{\Gamma}$ is said to be flexible if there exists an infinite number of such maps (upto equivalence by rigid transformations) and it is said to be movable if there exists an infinite number of injective maps. We study movability of Cayley graphs and construct regular moving graphs of all degrees. Further, we give explicit constructions of "dense", movable graphs.