On Digraphs Associated to Two Quadratic Forms Modulo n

Abstract

If n<∞ is a positive integer, R=Z_n is the ring of integers modulo n, with the assumption that on one hand G(Z_n) is a directed graph of the quadratic polynomial, x^2 +ax+b=0 mod(n), presented by the mapping φ_1:Z_n×Z_n→ Z_n×Z_n, defined as φ_1 (v_1,v_2 )=(v_1+v_2,v_1 v_2). On the other hand, Γ(Z_n) is a directed graph of the quadratic congruence x^2+c=0 mod(n), presented by the mapping φ_2:Z_n→ Z_n given by φ_2 (v)=v^2. We investigate the relationship between G and Γ, supporting the study with computer computations using Wolfram Mathematica software.