Journal of Prime Research in Mathematics

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

M. Haris Mateen
Department of mathematics, University of Punjab, Lahore, 54590, Pakistan.
M. Khalid Mahmood\(^1\)
Department of mathematics, University of Punjab Lahore, 54590, Pakistan.
\(^{1}\)Corresponding Author:


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.


Digraphs, Loops, Cycles, Components, Carmichael \(\lambda\)-function.