Digraphs associated with the kth power map on the quotient ring of polynomials over finite fields

Abstract

Let [Formula: see text] be a monic polynomial over a finite field [Formula: see text], and [Formula: see text] an integer. A digraph [Formula: see text] is one whose vertex set is [Formula: see text] and for which there is a directed edge from a polynomial [Formula: see text] to [Formula: see text] if [Formula: see text] in [Formula: see text]. If [Formula: see text], where the [Formula: see text]’s are distinct monic irreducible polynomials over [Formula: see text], then [Formula: see text] can be factorized as [Formula: see text]. In this work, we investigate the structure of these power digraphs. The semiregularity property is examined, and its relationship with the symmetric property is established. In addition, we look into the uniqueness of factorization of trees attached to a fixed point.