Sharp vertex connectivity of the Markoff graphs modulo $p$

Abstract

The Markoff graph $G_p$ modulo a prime $p$ is an undirected graph whose vertices are the nonzero solutions over the finite field ${\mathbb{F}}_p$ of the normalized Markoff equation \[ x_1^2+x_2^2+x_3^2=x_1x_2x_3, \] where two vertices are adjacent if they differ by a Vieta involution. 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 $\{G_p:p\ge 5\}$ forms an expander family. This motivates us to investigate the robustness of connectivity in the Markoff graphs. In this short note, we show 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 $p\ge 7$.