Investigating Metric Dimension and Edge Metric Dimension of Hexagonal Boron Nitride and Carbon Nanotubes

Abstract

When there is a difference in the distance between two vertices in a simple linked graph, then a vertex $x$ resolves both $u$ and $v$. If at least one vertex in $S$ distinguishes each pair of distinct vertices in $G$, then a set $S$ of vertices in $G$ is referred to as a resolving set. $G$'s metric dimension is the minimum number of vertices required in a resolving set. A subset $S$ of vertices in a simple connected graph is called an edge metric generator if each vertex can tell any two distinct edges $e_1$ and $e_2$ apart by their respective distances from each other. The edge metric dimension (EMD), denoted as $\mathrm{dim}_e(G)$, is the smallest cardinality of such a subset $S$ that serves as an edge metric generator for $G$. The primary objective of this study is to investigate the edge metric dimension (EMD) of hexagonal boron nitride and carbon nanotube structures.