The digraph of the kth power mapping of the quotient ring of polynomials over finite fields

Abstract

This work is based on ideas of Somer and Křížek on the digraphs associated with the congruence a k ≡ b mod n . We study the power digraph whose vertex set V f is the quotient ring A / f A and edge set is given by E f ( k ) = { ( g ¯ , g ¯ k ) : g ¯ ∈ A / f A } , where A = F q [ x ] , k > 1 and f ∈ A is a monic polynomial of degree ⩾1. Our main tool is the exponent of the unit group ( A / f A ) ⁎ and we obtain results on cycles and components parallel to those of Somer and of Křížek. This paper also generalizes the previous work on the digraph associated to the square mapping by the authors. In addition, we present some conditions when our digraphs are symmetric.