Computing the metric and partition dimension of H-Naphtalenic and VC₅C₇ nanotubes

It is well known that the relation between metric dimension and partition dimension of a non-trivial connected graph G denoted by dim(G) and pd(G) , respectively is given by the following inequality: pd(G) im(G) 1.However, the metric dimension of a connected graph G may be much larger than its partition dimension and this phenomena is called a 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 H-Naphtalenic and 5 7 VC C infinite nanotubes generated by tiling of the plane. We prove that the metric dimension of these two infinite nanotubes is not finite but their partition dimension is three, implying that these nanotubes are among the graphs having discrepancy between their metric dimension and partition dimension. It is natural to ask about characterization of the graphs having discrepancies between their metric dimension and partition dimension. Furthermore, it is also proved that there exist induced subgraphs of 2-dimensional lattices of these two nanotubes some of them have metric dimension depending upon n and others have constant metric dimension.