Geodesic
Graph rigidity properties of Ramanujan graphs
The electronic journal of combinatorics, Tome 30 (2023) no. 3
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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 $\mathbb{R}^2$.
DOI : 10.37236/11324
Classification : 05C50, 05C40, 52C25, 15A18
Mots-clés : eigenvalue, algebraic connectivity, connectivity, rigidity, redundant rigidity, global rigidity

Sebastian Cioabă    ; Sean Dewar  1   ; Georg Grasegger    ; Xiaofeng Gu 

1 RICAM 
@article{10_37236_11324,
     author = {Sebastian Cioab\u{a} and Sean Dewar and Georg Grasegger and Xiaofeng Gu},
     title = {Graph rigidity properties of {Ramanujan} graphs},
     journal = {The electronic journal of combinatorics},
     year = {2023},
     volume = {30},
     number = {3},
     doi = {10.37236/11324},
     zbl = {1519.05151},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/11324/}
}
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AU  - Georg Grasegger
AU  - Xiaofeng Gu
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DO  - 10.37236/11324
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Sebastian Cioabă; Sean Dewar; Georg Grasegger; Xiaofeng Gu. Graph rigidity properties of Ramanujan graphs. The electronic journal of combinatorics, Tome 30 (2023) no. 3. doi: 10.37236/11324

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