论文标题
$ r^2 $中图的光谱半径和（全球）刚度
Spectral radius and (globally) rigidity of graphs in $R^2$
论文作者
论文摘要
在过去的半个世纪中，$ r^2 $中图的刚度引起了极大的兴趣。 Lovász和Yemini（1982）证明，每$ 6 $连接的图形都在$ r^2 $中是刚性的。杰克逊和乔丹（Jackson andJordán，2005年）为$ r^2 $的全球刚度提供了类似的顶点连接性条件。这些结果表明，具有代数连接$μ（g）> 5 $的图形$ g $是（全球）刚度为$ r^2 $。 Cioabă，Dewar and Gu（2021）改进了这种界限，并证明了具有最低度$δ\ geq 6 $的图形$ g $在$ r^2 $中是刚性的，如果$ r^2 $如果$μ（g）> 2+ \ frac \ frac {1} {1} {δ-1} $，并且在$ r^2 $中是$ r^2 $的固定$μ（g）> 2+ \ frac {2} {δ-1} $。在本文中，我们从邻接特征值的角度研究了$ r^2 $的图形（全球）刚度。具体而言，我们为在光谱半径上的2连接（分别为3连接的）图提供了足够的条件，其最低度为刚性（全球刚度）。此外，我们确定了在所有最小刚性$ n $的最小刚性图中达到最大光谱半径的独特图。
Over the past half century, the rigidity of graphs in $R^2$ has aroused a great deal of interest. Lovász and Yemini (1982) proved that every $6$-connected graph is rigid in $R^2$. Jackson and Jordán (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 $μ(G)>5$ is (globally) rigid in $R^2$. Cioabă, Dewar and Gu (2021) improved this bound, and proved that a graph $G$ with minimum degree $δ\geq 6$ is rigid in $R^2$ if $μ(G)>2+\frac{1}{δ-1}$, and is globally rigid in $R^2$ if $μ(G)>2+\frac{2}{δ-1}$. 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 $n$.
