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.
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.