Graph rigidity properties of Ramanujan graphs

Abstract

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 $\mathbb{R}^2$.

Figures (9)

• Figure 1: The single special case to Theorem \ref{['thm:transitive']}. The graph is not globally rigid (see Theorem \ref{['t:jjs07']}), however it is rigid. To see that the graph is indeed rigid, note that if we delete a path of length 3 from each copy of $K_5$ contained in the graph, we obtain a $(2,3)$-tight graph.
• Figure 2: Two 4-regular vertex-transitive graphs that are not globally rigid in $\mathbb{R}^2$. The graph on the left is rigid in $\mathbb{R}^2$, but the graph on the right is not.
• Figure 3: (Left) A cubic Ramanujan graph with edge-connectivity one. (Right) A $4$-regular Ramanujan graph with edge-connectivity two
• Figure 4: The only $4$-regular Ramanujan graphs with at most 16 vertices that are not rigid in $\mathbb{R}^2$.
• Figure 5: $4$-regular bipartite Ramanujan graphs that are rigid but not globally rigid in $\mathbb{R}^2$.

Theorems & Definitions (47)

• Theorem 1.1: Cioabă, Dewar and Gu CDG21
• Theorem 1.2
• Theorem 1.3
• Theorem 2.1: Jackson and Jordán JaJo09
• proof : Proof of Theorem \ref{['thm:ramanujan']}
• Theorem 2.2: Jackson, Servatius and Servatius JSS07
• Lemma 2.3
• proof
• Theorem 2.4: Nilli nilli
• Lemma 2.5