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

摘要

This work is based on ideas of Somer and Kfizek 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) = Kg, g~k): g ∈ A/fA). 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/fA)* and we obtain results on cycles and components parallel to those of Somer and of Kfizek. 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.