Counting distinct functional graphs from linear finite dynamical systems

Let Fq be the finite field with q elements and, for each positive integer n, let Aq(n) be the number of non isomorphic func-tional graphs arising from Fq-linear maps T : Fqn -> Fqn. In 2013, Bach and Bridy proved that, for every prime power q, we have that 1 2 < lim inf loglogAq(n) log n < lim sup n ->+infinity n ->+infinity By combining some ideas from linear algebra, combinatorics and number theory, in this paper we provide sharper esti-mates on the function Aq(n) and, in particular, we prove that lim n ->+infinity log log Aq(n) log n < 1. log log Aq(n) log n = 1 for every prime power q.(c) 2022 Elsevier Inc. All rights reserved.