Enumeration of cyclic components of the power digraph Γ(n, 13)

Prabin Das, Pinkimani Goswami and Riazul Hoque
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 32, 2026, Number 3, Pages 556–575
DOI: 10.7546/nntdm.2026.32.3.556-575
Full paper (PDF, 296 Kb)

Details

Authors and affiliations

Prabin Das
Department of Mathematics, University of Science and Technology Meghalaya
Baridua, India

Pinkimani Goswami
Department of Mathematics, University of Science and Technology Meghalaya
Baridua, India

Riazul Hoque
Department of Mathematics, University of Science and Technology Meghalaya
Baridua, India

Abstract

For each positive integer n, we assign a digraph \Gamma(n,13) whose set of vertices is \mathbb{Z}_n=\lbrace 0,1,2, \ldots ,n-1\rbrace and attach exactly one directed edge from a vertex a to the vertex b if and only if a^{13}\equiv b \pmod n. In this work, the number of cycles and the number of components in the digraph \Gamma(n,13) are presented for n=p^k with the help of Carmichael’s lambda function. It is proved that for k\geq 1 and for the prime p=3, 5, 7 and 11, the respective numbers of components in the digraph \Gamma(p^k,13) are (4k-1), (4k+1), (18k-11) and (6k-1). The structure of tree associated with power digraph \Gamma(p^k, 13) for p=3, 5, 7 and 11 are also studied. We have also studied the structure of power digraph \Gamma(p, 13), where p=2^{2^m}+1, m\geq 0 is a Fermat prime.

Keywords

  • Carmichael λ-function
  • Components
  • Cycles
  • Digraph
  • Fermat prime
  • Fixed point
  • Power digraph

2020 Mathematics Subject Classification

  • 05C20
  • 11A07
  • 11A15

References

  1. Blanton Jr, E. L., Hurd, S. P., & McCranie, J. S. (1992). On a digraph defined by squaring modulo n. The Fibonacci Quarterly, 30(4), 322–334 .
  2. Bryant, S. (1967). Groups, graphs and Fermat’s last theorem. The American Mathematical Monthly, 74(2), 152–156.
  3. Carmichael, R. D. (1910). Note on a new number theory function. Bulletin of the American Mathematical Society, 16(5), 232–238.
  4. Goswami, P., Thakur, S. K., & Ray, G. C. (2023). The structure of power digraph connected with the congruence a11 ≡ b (mod n). Proyecciones (Antofagasta), 42(2), 457–477.
  5. Harary, F. (1969). Graph Theory. Addison-Wesley Publication Company, London.
  6. Křížek, M., Luca, F., & Somer, L. (2001). 17 Lectures on Fermat Numbers: From Number Theory to Geometry. CMS Books in Mathematics, Vol. 9, Springer-Verlag, New York.
  7. Křížek, M., & Somer, L. (2004). Sophie Germain little suns. Mathematica Slovaca, 54(5), 433–442.
  8. Lucheta, C., Miller, E., & Reiter, C. (1996). Digraphs from powers modulo p. The Fibonacci Quarterly, 34, 226–239.
  9. Mateen, M. H., & Mahmood, M. K. (2019). Power digraphs associated with the congruence xn ≡ y (mod m). Punjab University Journal of Mathematics, 51(5), 93–102.
  10. Rahmati, M. (2017). Some digraphs attached with the congruence x5 ≡ y (mod n). Journal of Mathematical Extension, 11(1), 47–56.
  11. Rogers, T. D. (1996). The graph of the square mapping on the prime fields. Discrete Mathematics, 148, 317–324.
  12. Sawkmie, A., & Singh, M. M. (2015). On the tree structure of the power digraphs modulo n. Czechoslovak Mathematical Journal, 65(4), 923–945.
  13. Sawkmie, A., & Singh, M. M. (2018). On the uniqueness of the factorization of power digraphs modulo n. Rendiconti del Seminario Matematico della Universita di Padova, 140, 185–219.
  14. Skowronek-Kaziów, J. (2009). Properties of digraphs connected with some congruence relations. Czechoslovak Mathematical Journal, 59, 39–49.
  15. Somer, L., & Křížek, M. (2004). On a connection of number theory with graph theory. Czechoslovak Mathematical Journal, 54, 465–485.
  16. Somer, L., & Křížek, M. (2009). On symmetric digraphs of the congruence xk ≡ y (mod n). Discrete Mathematics, 309, 1999–2009.
  17. Somer, L., & Křížek, M. (2011). The structure of digraphs associated with the congruence xk ≡ y (mod n). Czechoslovak Mathematical Journal, 61, 337–358.
  18. Szalay, L. (1992). Egy diszkrét iteráció a számelméletben. Berzsenyi Dániel Tanárképző Főiskola Tudományos Közleményei, 8, 71–91 (in Hungarian).
  19. Thakur, S. K., Goswami, P., & Ray, G. C. (2023). Enumeration of cyclic vertices and components over the congruence a11 ≡ b (mod n). Notes on Number Theory and Discrete Mathematics, 29(3), 525–537.

Manuscript history

  • Received: 7 October 2025
  • Revised: 19 June 2026
  • Accepted: 16 October 2026
  • Online First: 23 October 2026

Copyright information

Ⓒ 2026 by the Authors.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Related papers

Cite this paper

Das, P., Goswami, P., & Hoque, R. (2026). Enumeration of cyclic components of the power digraph Γ(n, 13). Notes on Number Theory and Discrete Mathematics, 32(3), 556-575, DOI: 10.7546/nntdm.2026.32.3.556-575.

Comments are closed.