Journal of Prime Research in Mathematics

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


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.


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