Graph rigidity properties of Ramanujan graphs

TL;DR

This paper investigates the global rigidity of Ramanujan graphs in the plane, extending known results to new degrees and graph classes, and includes computational analysis of small cases.

Contribution

It extends global rigidity results for Ramanujan graphs to degrees 4, 5, 6, 7, and large orders, and explores rigidity on various surfaces and frameworks.

Findings

Ramanujan graphs with degree ≥8 are globally rigid in .

Large Ramanujan graphs with degrees 6 and 7 are globally rigid.

Certain small Ramanujan graphs are computationally verified for rigidity.

Abstract

A recent result of Cioab\u{a}, Dewar and Gu implies that any $k$-regular Ramanujan graph with $k\geq 8$ is globally rigid in $\mathbb{R}^2$. In this paper, we extend these results and prove that any $k$-regular Ramanujan graph of sufficiently large order is globally rigid in $\mathbb{R}^2$ when $k\in \{6, 7\}$, and when $k\in \{4,5\}$ if it is also vertex-transitive. These results imply that the Ramanujan graphs constructed by Morgenstern in 1994 are globally rigid. We also prove several results on other types of framework rigidity, including body-bar rigidity, body-hinge rigidity, and rigidity on surfaces of revolution. In addition, we use computational methods to determine which Ramanujan graphs of small order are globally rigid in $\mathbb{R}^2$.