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

摘要

Let f be a monic polynomial over a finite field F-q, and k >= 2 an integer. A digraph G( f, k) is one whose vertex set is F-q[x]/< f > and for which there is a directed edge from a polynomial g(1) to g(2) if g(1)(k) = g(2) in F-q[x]/< f >. If f = P-1(e1) P-2(e2) . . . P-r(er), where the Pi's are distinct F-q, then G(f, k) can be factorized as G(f, k) congruent to G(P-1(e1), k) x G(P-2(e2) ,k) x . . . x G(P-r(er) ,k). 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.