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

Abstract

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 $ |G| $, then the diameters of all nondiagonal orbital graphs of $ VG $ are at most $ 9d^3 $. This improves an earlier exponential bound by A. Mar\'oti and the author.