模p的马尔可夫图的Sharp顶点连通性
Sharp vertex connectivity of the Markoff graphs modulo $p$
- University of Science and Technology of China（中国科学技术大学）
- Tsinghua University（清华大学）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
该研究针对模p的马尔可夫图G_p，证明若其连通则必为2-连通，且对所有足够大的素数p成立，同时指出其非3-连通的最优性，回应了该序列是否为扩张器族的相关问题。
中文摘要 AI 辅助
模素数p的马尔可夫图G_p是一个无向图，其顶点是标准化马尔可夫方程x₁²+x₂²+x₃²=x₁x₂x₃在有限域F_p上的非零解，若两个顶点通过维塔对合变换相连，则它们相邻。Bourgain、Gamburd和Sarnak的重大突破证明G_p包含一个巨型连通分量，结合Chen的卓越可除性定理，这意味着对于所有足够大的素数p，G_p是连通的。在同一篇论文中，Bourgain、Gamburd和Sarnak进一步提出，序列{G_p: p≥5}是否构成扩张器族，这促使我们研究马尔可夫图连通性的鲁棒性。在本短文中，我们证明若马尔可夫图G_p是连通的，则它实际上是2-连通的，因此对于所有足够大的素数p，马尔可夫图G_p都是2-连通的；该结论是最优的，因为对于任何素数p≥7，G_p都不是3-连通的。
英文摘要
For a prime $p$ and $k\in\mathbb{F}_p$, the generalized Markoff graph $G_{p,k}$ is an undirected graph whose vertices are the solutions over the finite field $\mathbb{F}_p$ of the normalized Markoff equation \[ x_1^2+x_2^2+x_3^2=x_1x_2x_3+k, \] where two vertices are adjacent if they differ by a Vieta involution. The Markoff graph $G_p$ is obtained from $G_{p,0}$ by removing the origin. The structure of $G_p$ has been the subject of extensive study; in particular, a major breakthrough of Bourgain, Gamburd, and Sarnak established that $G_p$ contains a giant connected component. Combined with Chen's remarkable divisibility theorem, this implies that $G_p$ is connected for all sufficiently large primes $p$. In the same paper, Bourgain, Gamburd, and Sarnak further asked whether the family $\{G_p\colon \text{primes }p\geq 5\}$ forms an expander family. This motivates us to investigate the robustness of connectivity in the Markoff graphs. The main result of this paper is proved in the general setting: for every prime $p\geq5$ and every $k\in\mathbb{F}_p\setminus\{4\}$, each connected component $C$ of $G_{p,k}$ with $|V(C)|\geq 3$ is $2$-connected. Reducing to the case $k=0$, we conclude that if the Markoff graph $G_p$ is connected, then it is in fact $2$-connected. Consequently, the Markoff graph $G_p$ is $2$-connected for all sufficiently large primes $p$. This is sharp in the sense that $G_p$ is not $3$-connected for any prime $p\geq 7$.
