6 problems

Let $p$ be a prime, and let $I_p$ denote the number of parameters $a$ for which the functional graph associated with the quadratic polynomial $f_a$ is connected. The computations u…

Let $G$ be a finite group of order $n$, let $C_n$ denote the cyclic group of order $n$, and let $N(a,G)$ be the number of connected components of the functional graph $G(a,G)$ asso…

The one-node tree conjecture.

Let $p\geq3$ be prime. Let ${\mathscr A}_p$ and ${\mathscr B}_p^*$ be the parameter sets defined in the paper, with ${\mathscr A}_p$ denoting the relevant set of parameters and…

Let $p$ be prime, and let $I_p$ denote the number of parameters $a\in\mathbb F_p$ for which the functional graph ${\mathcal G}_a$ of $f_a(X)=X^2+a$ is connected. The existence conj…

Polynomial cycle-count conjecture. For any $k\in\mathbb{Z}^{>0}$ and any $d\in\mathbb{Z}^{\geq 0}$,