Computing metric and partition dimension of 2-dimensional lattices of certain nanotubes

The metric dimension (location number) of a connected graph G denoted by β(G) and partition dimension of a connected graph G denoted by pd(G) are related as pd(G) ≤ β(G)+1. However, the metric dimension may be much larger than the partition dimension and this phenomena is known as discrepancy between metric dimension and partition dimension. In this paper, we study the metric dimension (location number) and partition dimension of 2-dimensional lattices of certain infinite nanotubes generated by the tiling of the plane. We prove that the metric dimension of these nanotubes is not finite but their partition dimension is finite and evaluated, implying that these nanostructures are among the graphs having discrepancy between their metric dimension and partition dimension. It is natural to think about the characterization of the graphs having discrepancies between their metric dimension and partition dimension. It is also proved that there exist induced subgraphs of 2-dimensional lattices of these nanostructures having unbounded metric dimension as well as having constant metric dimension.