The quadratic digraph on polynomial rings over finite fields

Abstract

This work is based on ideas of Somer and of Křížek on the structure of digraphs associated with quadratic congruence modulo n. We study the quadratic digraph whose vertex set V f is the quotient ring A / f A and edge set is given by E f ( 2 ) = { ( g ¯ , g ¯ 2 ) : g ¯ ∈ A / f A } , where A = F q [ x ] and f ∈ A is a monic polynomial of degree ⩾1 in A. Our main tool is the exponent of the unit group ( A / f A ) ∗ and we obtain results parallel to them.