A new approach for the enumeration of components of Digraphs over quadratic maps

Various partial attempts to count cycles and components of digraphs from congruences have been made earlier. While the problem is still open till date. In this work, we introduce a new approach to solve the problem over quadratic congruence equations. Define a mapping \(g:Z_{m}\mapsto Z_{m}\) by \(g(t)=t^{2}\), where \(Z_{m}\) is the ring of residue classes modulo \(m\). The digraph \(G(2,m)\) over the set of residue classes assumes an edge between the residue classes \(\overline{x}\) and \(\overline{y}\) if and only if \(g(\overline{x})\equiv \overline{y}~(\text{mod}~m)\) for \(m\in Z^{+}\). Classifications of cyclic and non-cyclic vertices are proposed and proved using basic modular arithmetic. Finally, explicit formulas for the enumeration of non-isomorphic components are proposed followed by simple proofs from number theory.