**Sanjay Kumar Thakur, Pinkimani Goswami and Gautam Chandra Ray**

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 29, 2023, Number 3, Pages 525–537

DOI: 10.7546/nntdm.2023.29.3.525-537

**Full paper (PDF, 3.6 Mb)**

## Details

### Authors and affiliations

Sanjay Kumar Thakur

*Department of Mathematics, Science College Kokrajhar
Assam, India
*

Pinkimani Goswami

*Department of Mathematics, University of Science and Technology Meghalaya
Baridua, India
*

Gautam Chandra Ray

*Department of Mathematics, CIT Kokrajhar
Assam, India
*

### Abstract

For each positive integer , we assign a digraph whose set of vertices is and there exists exactly one directed edge from the vertex to the vertex iff . Using the ideas of modular arithmetic, cyclic vertices are presented and established for in the digraph . Also, the number of cycles and the number of components in the digraph is presented for with the help of Carmichael’s lambda function. It is proved that for , the number of components in the digraph is and for the digraph has non-isomorphic cycles of length greater than , whereas the number of components of the digraph is .

### Keywords

- Digraph
- Fixed point
- Power digraph
- Carmichael λ-function
- Cycles
- Components

### 2020 Mathematics Subject Classification

- 05C20
- 11A07
- 11A15

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### Manuscript history

- Received: 25 August 2022
- Revised: 3 March 2023
- Accepted: 20 July 2023
- Online First: 26 July 2023

### Copyright information

Ⓒ 2023 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).

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## Cite this paper

Thakur, S. K., Goswami, P., & Ray, G. C. (2023). Enumeration of cyclic vertices and components over the congruence . *Notes on Number Theory and Discrete Mathematics*, 29(3), 525-537, DOI: 10.7546/nntdm.2023.29.3.525-537.