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: khalid.math@pu.edu.pk

### Abstract

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.

#### Keywords:

Digraphs, Loops, Cycles, Components, Carmichael $$\lambda$$-function.