On a polynomial bound for the orbital diameter of primitive affine groups

Let $VG$ be a finite primitive affine permutation group, where $V$ is a vector space of dimension $d$ over the prime field ${\mathbb{F}}_p$ and $G$ is an irreducible linear group on $V$. We prove that if $p$ divides $\mathrm{\mid }G\mathrm{\mid }$, then the diameters of all nondiagonal orbital graphs of $VG$ are at most $9d^3$. This improves an earlier exponential bound by A. Maróti and the author.