A Classification of Cyclic Nodes and Enumeration of Components of a Class of Discrete Graphs

Authors

M. Khalid Mahmood, Department of Mathematics, University of the Punjab, Lahore, PakistanFollow
Farooq Ahmad, Faculty of Information Technology, University of Central Punjab, Lahore, PakistanFollow

Author Country (or Countries)

Pakistan

Abstract

Let Zn be the ring of residue classes modulo n. Define f : Zn 7→ Zn by f (x) = x4. Action of this map is studied by means of digraphs which produce an edge from the residue classes a to b if f (a) ≡ b. For every integer n, an explicit formula is given for the number of fixed points of f . It is shown that the graph G(pk), k ≥ 1 has four fixed points if and only if 3 | p−1 and has two fixed points if and only if 3 ∤ p−1. A classification of cyclic vertices of the graph G(pk) has been determined. A complete enumeration of non-isomorphic cycles and components of G(pk) has been explored.

Digital Object Identifier (DOI)

http://dx.doi.org/10.12785/amis/090115

Recommended Citation

Khalid Mahmood, M. and Ahmad, Farooq (2023) "A Classification of Cyclic Nodes and Enumeration of Components of a Class of Discrete Graphs," Applied Mathematics & Information Sciences: Vol. 09: Iss. 1, Article 22.
DOI: http://dx.doi.org/10.12785/amis/090115
Available at: https://digitalcommons.aaru.edu.jo/amis/vol09/iss1/22