Abstract:

A linear transformation [math]A: (\mathbb{F}_q)^n \to (\mathbb{F}_q)^n[/math] gives rise to a directed graph by regarding the elements of [math](\mathbb{F}_q)^n[/math] as vertices and drawing an edge from [math]v[/math] to [math]w[/math] if [math]Av = w[/math]. In 1959, Elspas determined the "functional graphs" on [math]q^n[/math] vertices that are realized in this way. In doing so he showed that there are many nonsimilar linear transformations which have isomorphic functional graphs (and so are conjugate by a nonlinear permutation of [math](\mathbb{F}_q)^n)[/math]. I review some of this work and prove an new upper bound on the number of equivalence classes of affinelinear transformations of [math](F_q)^n[/math] under the equivalence relation of isomorphism of functional graphs. This bound is significantly smaller than the number of conjugacy classes of [math]\operatorname{GL}_n(q)[/math]. This is joint work with Eric Bach.
