Graph rigidity properties of Ramanujan graphs

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 $\mathbb{R}^2$.