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arXiv · 2206.03983

Graph rigidity properties of Ramanujan graphs

Sebastian M. Cioabă·Sean Dewar·Georg Grasegger·Xiaofeng Gu
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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 $\mathbb{R}^2$.

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Sebastian M. Cioabă, Sean Dewar, Georg Grasegger, Xiaofeng Gu. 2022-06-08. Graph rigidity properties of Ramanujan graphs. https://doi.org/10.37236/11324

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