Journal of Prime Research in Mathematics

# 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.
Sara Shoukat
Center for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya
University, Multan, Pakistan.

### 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.