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

This work is based on ideas of Somer and Krizek on the digraphs associated with the congruence a(k) equivalent to b mod n. We study the power digraph whose vertex set V(f) is the quotient ring A/fA and edge set is given by E(f)((k)) = {((g) over bar.(g) over bar (k)): (g) over bar is an element of A/fA}, where A = F(q)inverted right perpendicularxinverted left perpendicular, k > 1 and f is an element of 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 Krizek. 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. (C) 2011 Elsevier Inc. All rights reserved.