**Journal of Prime Research in Mathematics**

Vol. 1 (2008), Issue 1, pp. 154 – 164

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

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

# On the partition dimension of some wheel related graphs

**Imran Javaid
**Center for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya

University, Multan, Pakistan.

**Center for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya**

Sara Shoukat

Sara Shoukat

University, Multan, Pakistan.

Copyright © 2008 Imran Javaid, Sara Shoukat**. **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:**December, 2008.

### Abstract

Let G be a connected graph. For a vertex \(v ∈ V (G)\) and an ordered \(k-\)partition \(Π = {S_1, S_2, …, S_k}\) of \(V (G)\), the representation of \(v\) with respect to \(Π\) is the \(k-\)vector r \((v|Π) = (d(v, S_1), d(v, S_2), …, d(v, S_k))\) where \(d(v, S_i) = min_{w∈S_i} d(v, w)(1 ≤ i ≤ k)\). The k-partition \(Π\) is said to be resolving if the k-vectors \(r(v|Π), v ∈ V (G)\), are distinct. The minimum \(k\) for which there is a resolving \(k\)-partition of \(V (G)\) is called the partition dimension of \(G\), denoted by \(pd(G)\). In this paper, we give upper bounds for the cardinality of vertices in some wheel related graphs namely gear graph, helm, sunflower and friendship graph with given partition dimension \(k\).

#### Keywords:

Resolving partition, partition dimension, gear graph, helm, sunflower and friendship graph.