Spectral radius and (globally) rigidity of graphs in $R^2$

TL;DR

This paper explores the relationship between spectral properties of graphs, specifically spectral radius, and their rigidity in the plane, providing new spectral conditions for rigidity and identifying extremal graphs.

Contribution

It introduces spectral radius-based criteria for graph rigidity and globally rigidity in the plane, extending previous connectivity and algebraic connectivity results.

Findings

Spectral radius conditions guarantee rigidity and global rigidity in certain connectivity classes.

Identifies the unique extremal graph with maximum spectral radius among minimally rigid graphs.

Provides bounds linking spectral radius and graph rigidity properties.

Abstract

Over the past half century, the rigidity of graphs in $R^2$ has aroused a great deal of interest. Lov\'{a}sz and Yemini (1982) proved that every $6$-connected graph is rigid in $R^2$. Jackson and Jord\'{a}n (2005) provided a similar vertex-connectivity condition for the globally rigidity of graphs in $R^2$. These results imply that a graph $G$ with algebraic connectivity $\mu(G)>5$ is (globally) rigid in $R^2$. Cioab\u{a}, Dewar and Gu (2021) improved this bound, and proved that a graph $G$ with minimum degree $\delta\geq 6$ is rigid in $R^2$ if $\mu(G)>2+\frac{1}{\delta-1}$, and is globally rigid in $R^2$ if $\mu(G)>2+\frac{2}{\delta-1}$. In this paper, we study the (globally) rigidity of graphs in $R^2$ from the viewpoint of adjacency eigenvalues. Specifically, we provide sufficient conditions for a 2-connected (resp. 3-connected) graph with given minimum degree to be rigid (resp. globally rigid) in terms of the spectral radius. Furthermore, we determine the unique graph attaining the maximum spectral radius among all minimally rigid graphs of order $n$.