The Wiener, hyper-Wiener, Harary and SK indices of the P(Zpk·qr) power graph

Volkan Aşkin
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 29, 2023, Number 4, Pages 794–803
DOI: 10.7546/nntdm.2023.29.4.794-803
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Volkan Aşkin
Department of Mathematics, University of Gazi
‎06500, Ankara, Turkey

Abstract

The undirected P(Z_n) power graph of a finite group of Z_n is a connected graph, the set of vertices of which is Z_n. Here \langle u, v\rangle \in P(Z_n) are two diverse adjacent vertices if and only if u \ne v and \langle v \rangle \subseteq \langle u \rangle or \langle u \rangle \subseteq \langle v \rangle. We will shortly name the undirected P(Z_n) power graph as the power graph P(Z_n). The Wiener, hyper-Wiener, Harary and SK indices of the P(Z_n) power graph are in order as follows

    \[\frac{1}{2}\underset{\left\{ u,v \right\}\subseteq V\left( G \right)}{\mathop \sum }\,d\left( u,v \right), \ \frac{1}{2}\underset{\left\{ u,v \right\}\subseteq V\left( G \right)}{\mathop \sum }\,d\left( u,v \right)+\frac{1}{2}\underset{\left\{ u,v \right\}\subseteq V\left( G \right)}{\mathop \sum }\,{{d}^{2}}\left(u,v \right),\]

    \[\underset{\left\{ u,v \right\}\subseteq V\left( G \right)}{\mathop \sum }\,\frac{1}{d\left(u,v \right)} \mbox{ and } \frac{1}{2}\underset{uv\in E\left( G \right)}{\mathop \sum }\,\left( {{d}_{u}}+{{d}_{v}} \right).\]

In this article we focus more on the indices of P(Z_n) power graph by Wiener, hyper-Wiener, Harary and SK the definition of the power graph is presented and the results and theorems which we need in our discussion are provided in the introduction. Finally, the main point of the article is that we calculate the Wiener, hyper-Wiener, Harary and SK indices of the power graph P(Z_n) corresponding to the vertex n = p^k \cdot q^r. These are as follows: p, q are distinct primes and k, r are nonnegative integers.

Keywords

  • Wiener index
  • Hyper-Wiener index
  • Harary index
  • SK index
  • Undirected power graph

2020 Mathematics Subject Classification

  • 05C09

References

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Manuscript history

  • Received: 9 March 2023
  • Revised: 30 October 2023
  • Accepted: 24 November 2023
  • Online First: 30 November 2023

Copyright information

Ⓒ 2023 by the Author.
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

Aşkin, V. (2023). The Wiener, hyper-Wiener, Harary and SK indices of the P(Zpk·qr) power graph. Notes on Number Theory and Discrete Mathematics, 29(4), 794-803, DOI: 10.7546/nntdm.2023.29.4.794-803.

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