Connectivity of Markoff mod-p graphs and maximal divisors

Markoff mod-$p$ graphs are conjectured to be connected for all primes $p$. In this paper, we use results of Chen and Bourgain, Gamburd, and Sarnak to confirm the conjecture for all $p > 3.448\cdot10^{392}$. We also provide a method that quickly verifies connectivity for many primes below this bound. In our study of Markoff mod-$p$ graphs we introduce the notion of \emph{maximal divisors} of a number. We prove sharp asymptotic and explicit upper bounds on the number of maximal divisors, which ultimately improves the Markoff graph $p$-bound by roughly 140 orders of magnitude as compared with an approach using all divisors.