AI 中文总结
本文研究伪随机图的刚性性质，证明满足一定条件的C-扩张图和满足λ≤c₁r的(n,r,λ)-图分别具有指定阶数的刚性，结果在涉及的通用常数范围内是最优的，改进并拓展了此前相关研究。
AI 中文摘要
若图$G=(V,E)$的顶点在$\boldsymbol{\text{R}}^d$中处于一般嵌入状态时，所有保持相邻顶点间距离不变的顶点连续运动，均由$\boldsymbol{\text{R}}^d$的等距变换（即整个图的平移与旋转）诱导，则称该图为$d$-刚性图。本文研究伪随机图的刚性性质。首先，考虑$C$-扩张图，这是一类近期在伪随机图哈密顿性研究中被探讨的图类，其定义为：$n$个顶点的图中，每个大小小于$n/(2C)$的顶点集$A$，其邻域大小至少为$C|A|$；且对每对大小至少为$n/(2C)$的不相交顶点集$A,B$，$A$与$B$之间至少存在一条边。我们证明，对所有$C\boldsymbol{\text{≥}}8$且整数$n\boldsymbol{\text{≥}}9C$，每个$n$顶点的$C$-扩张图都是$\boldsymbol{\text{⌊}}C/8\boldsymbol{\text{⌋}}$-刚性图。其次，研究$(n,r,\boldsymbol{\text{λ}})$-图，即$n$个顶点的$r$-正则图，其非平凡邻接特征值的绝对值均不超过$\boldsymbol{\text{λ}}$，这是一类知名的图族，具备多种伪随机性质。我们证明存在绝对常数$c_1,c_2>0$，使得每个满足$\boldsymbol{\text{λ}}\boldsymbol{\text{≤}}c_1r$的$(n,r,\boldsymbol{\text{λ}})$-图$G$都是$\boldsymbol{\text{⌊}}c_2r\boldsymbol{\text{⌋}}$-刚性图。本文结果在涉及的通用常数取值范围内是最优的，且改进并拓展了作者此前关于随机图与伪随机图刚性的研究工作。
英文摘要
A graph $G=(V,E)$ is called $d$-rigid if, for a generic embedding of its vertices in $\mathbb{R}^d$, the only continuous motions of the vertices preserving the distances between all pairs of adjacent vertices are those induced from the isometries of $\mathbb{R}^d$ (that is, translations and rotations of the whole graph). In this paper, we study rigidity properties of pseudorandom graphs. First, we consider $C$-expander graphs, a class of graphs recently studied in the context of Hamiltonicity of pseudorandom graphs. These are $n$-vertex graphs for which every vertex set $A$ of size smaller than $n/(2C)$ has a neighbourhood of size at least $C|A|$, and for every pair of disjoint sets $A,B$ of size at least $n/(2C)$ each, there is at least one edge between $A$ and $B$. We show that for every $C\ge 8$ and every integer $n\ge 9C$, every $n$-vertex $C$-expander is $\lfloor C/8\rfloor$-rigid. Next, we study $(n,r,λ)$-graphs, which are $n$-vertex $r$-regular graphs whose non-trivial adjacency eigenvalues are bounded in absolute value by $λ$. This is a well-known family of graphs, known to possess various pseudorandom properties. We prove that there exist absolute constants $c_1,c_2>0$ such that every $(n,r,λ)$-graph $G$ with $λ\le c_1r$ is $\lfloor c_2r\rfloor$-rigid. Our results are sharp up to the value of the universal constants involved, and they improve and extend previous work by the authors on the rigidity of random and pseudorandom graphs.
Comments13 pages
