Structure of digraphs associated with quadratic congruences with composite moduli

Abstract

We assign to each positive integer n a digraph G ( n ) whose set of vertices is H = { 0 , 1 , … , n - 1 } and for which there exists a directed edge from a ∈ H to b ∈ H if a 2 ≡ b ( mod n ) . Associated with G ( n ) are two disjoint subdigraphs: G 1 ( n ) and G 2 ( n ) whose union is G ( n ) . The vertices of G 1 ( n ) correspond to those residues which are relatively prime to n. The structure of G 1 ( n ) is well understood. In this paper, we investigate in detail the structure of G 2 ( n ) .