[Paper Review] Spectral conditions for graph rigidity in the Euclidean plane

Sebastian M. Cioabă, Dewar, Sean+1 more|arXiv (Cornell University)|Jan 20, 2020

Structural Analysis and Optimization29 references4 citations

TL;DR

This paper establishes improved spectral conditions for graph rigidity and global rigidity in the Euclidean plane using algebraic connectivity (the second smallest Laplacian eigenvalue). It proves that if a graph has minimum degree $\delta \geq 6$ and algebraic connectivity greater than $2 + \frac{1}{\delta-1}$, it is rigid; if greater than $2 + \frac{2}{\delta-1}$, it is globally rigid. These results extend to regular Ramanujan graphs with degree at least 8, which are globally rigid.

ABSTRACT

Rigidity is the property of a structure that does not flex. It is well studied in discrete geometry and mechanics, and has applications in material science, engineering and biological sciences. A bar-and-joint framework is a pair $(G,p)$ of graph $G$ together with a map $p$ of the vertices of $G$ into the Euclidean plane. We view the edges of $(G, p)$ as bars and the vertices as universal joints. The vertices can move continuously as long as the distances between pairs of adjacent vertices are preserved. The framework is rigid if any such motion preserves the distances between all pairs of vertices. In 1970, Laman obtained a combinatorial characterization of rigid graphs in the Euclidean plane. In 1982, Lovász and Yemini discovered a new characterization and proved that every $6$-connected graph is rigid. Combined with a characterization of global rigidity, their proof actually implies that every 6-connected graph is globally rigid. Consequently, if Fiedler's algebraic connectivity is greater than 5, then $G$ is globally rigid. In this paper, we improve this bound and show that for a graph $G$ with minimum degree $δ\geq 6$, if its algebraic connectivity is greater than $2+\frac{1}{δ-1}$, then $G$ is rigid and if its algebraic connectivity is greater than $2+\frac{2}{δ-1}$, then $G$ is globally rigid. Our results imply that every connected regular Ramanujan graph with degree at least $8$ is globally rigid. We also prove a more general result giving a sufficient spectral condition for the existence of $k$ edge-disjoint spanning rigid subgraphs. The same condition implies that a graph contains $k$ edge-disjoint spanning $2$-connected subgraphs. This result extends previous spectral conditions for packing edge-disjoint spanning trees.

Motivation & Objective

• To improve existing spectral conditions for rigidity and global rigidity in 2D bar-and-joint frameworks.
• To establish tighter bounds on algebraic connectivity that guarantee rigidity and global rigidity.
• To extend spectral conditions beyond edge-disjoint spanning trees to include edge-disjoint spanning 2-connected subgraphs and rigid subgraphs.
• To analyze the rigidity of pseudo-random and Ramanujan graphs using spectral graph theory.
• To provide a spectral analogue of combinatorial results on graph packing and connectivity.

Proposed method

• The authors use the second smallest eigenvalue of the Laplacian matrix, $\mu_2(G)$, as a spectral measure of connectivity.
• They derive sufficient conditions on $\mu_2(G)$ in terms of minimum degree $\delta$ to ensure rigidity and global rigidity.
• The analysis builds on Laman’s combinatorial characterization of rigid graphs and Lovász–Yemini’s result that 6-connected graphs are globally rigid.
• The paper applies Courant–Weyl inequalities to relate $\mu_2$ to eigenvalues of the adjacency and signless Laplacian matrices.
• It generalizes results to $k$ edge-disjoint spanning rigid and 2-connected subgraphs via spectral thresholds.
• The framework is applied to Ramanujan graphs, showing that $d$-regular Ramanujan graphs with $d \geq 8$ are globally rigid.

Experimental results

Research questions

• RQ1 What spectral condition on the algebraic connectivity ensures that a graph is rigid in the Euclidean plane?
• RQ2 Can the bound of 6-connectivity for global rigidity be improved using spectral graph theory?
• RQ3 What is the minimal algebraic connectivity threshold that guarantees global rigidity for graphs with minimum degree $\delta \geq 6$?
• RQ4 How can spectral conditions be extended to guarantee $k$ edge-disjoint spanning rigid or 2-connected subgraphs?
• RQ5 Do Ramanujan graphs with degree $d \geq 8$ satisfy the new spectral condition for global rigidity?

Key findings

• For a graph with minimum degree $\delta \geq 6$, if its algebraic connectivity exceeds $2 + \frac{1}{\delta-1}$, then the graph is rigid in $\mathbb{R}^2$.
• If the algebraic connectivity exceeds $2 + \frac{2}{\delta-1}$, then the graph is globally rigid in $\mathbb{R}^2$.
• Every connected $d$-regular Ramanujan graph with $d \geq 8$ is globally rigid in $\mathbb{R}^2$.
• The spectral condition $\mu_2(G) > 2 + \frac{2k-1}{\delta-1}$ implies the existence of $k$ edge-disjoint spanning 2-connected subgraphs.
• The results extend previous spectral conditions for edge-disjoint spanning trees to include edge-disjoint spanning 2-connected subgraphs.
• The same spectral threshold implies that a graph contains $k$ edge-disjoint spanning rigid subgraphs.