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Let $\mathbb F_q$ be the finite field with $q$ elements and, for eachpositive integer $n$, let $A_q(n)$ be the number of non isomorphic functionalgraphs arising from $\mathbb F_q$-linear maps $T:\mathbb F_{q}^n\to \mathbbF_{q}^n$. In 2013, Bach and Bridy proved that, if $q$ is fixed and $n$ issufficiently large, the quantity $\frac{\log \log A_q(n)}{\log n}$ lies in theinterval $[\frac{1}{2}, 1]$. By combining some ideas from linear algebra,combinatorics and number theory, in this paper we provide sharper estimates onthe function $A_q(n)$ and, in particular, we prove that $\lim\limits_{n\to+\infty}\frac{\log\log A_q(n)}{\log n}=1$ for every prime power $q$.

Counting distinct functional graphs from linear finite dynamical systems