**Journal of Prime Research in Mathematics**

Vol. 17 (2021), Issue 1, pp. 70 – 83

ISSN: 1817-3462E (Online) 1818-5495 (Print)

ISSN: 1817-3462E (Online) 1818-5495 (Print)

# Partition dimension of generalized Peterson and Harary graphs

**Abdul Jalil M. Khalaf\(^a\), Muhammad Faisal Nadeem\(^{b,*}\), Muhammasd Azeem\(^c\), Mohammad Reza Farahani\(^d\), Murat Cancann\(^e\)**

\(^a\)Department of Mathematics, Faculty of Computer Science and Mathematics University of Kufa, Najaf, Iraq.

\(^b\)Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore 54000, Pakistan.

\(^c\)Department of Aerospace Engineering, Faculty of Engineering, Universiti Putra Malaysia, Malaysia.

\(^d\)Department of Mathematics, Iran University of Science and Technology Narmak, Tehran, Iran.

\(^e\)Faculty of Education, Van Yüzüncü Yil University, Van, Turkey.

Correspondence should be addressed to: Muhammad Faisal Nadeem at mfaisalnadeem@ymail.com

Copyright © 2021 Abdul Jalil M. Khalaf, Muhammad Faisal Nadeem, Muhammasd Azeem, Mohammad Reza Farahani, Murat Cancan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Published:**Received: 5 January 2021; Accepted: 25 May 2021; Published Online: 30 June 2021.

### Abstract

The distance of a connected, simple graph \(\mathbb{P}\) is denoted by \(d({\alpha}_1,{\alpha}_2),\) which is the length of a shortest path between the vertices \({\alpha}_1,{\alpha}_2\in V(\mathbb{P}),\) where \(V(\mathbb{P})\) is the vertex set of \(\mathbb{P}.\) The \(l\)-ordered partition of \(V(\mathbb{P})\) is \(K=\{K_1,K_2,\dots,K_l\}.\) A vertex \({\alpha}\in V(\mathbb{P}),\) and \(r({\alpha}|K)=\{d({\alpha},K_1),d({\alpha},K_2),\dots,d({\alpha},K_l)\}\) be a \(l\)-tuple distances, where \(r({\alpha}|K)\) is the representation of a vertex \({\alpha}\) with respect to set \(K.\) If \(r({\alpha}|K)\) of \({\alpha}\) is unique, for every pair of vertices, then \(K\) is the resolving partition set of \(V(\mathbb{P}).\) The minimum number \(l\) in the resolving partition set \(K\) is known as partition dimension (\(pd(\mathbb{P})\)). In this paper, we studied the generalized families of Peterson graph, \(P_{{\lambda},{\chi}}\) and proved that these families have bounded partition dimension.

#### Keywords:

Generalized Peterson graph, Harary Graph, partition dimension, partition resolving set, sharp bounds of partition dimension.