Graph Rigidity Properties of Ramanujan Graphs

  • Sebastian M. Cioabă
  • Sean Dewar
  • Georg Grasegger
  • Xiaofeng Gu


A recent result of Cioabă, 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 $\mathbb{R}^2$.

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