Exploring the metric dimensions of graphene: Unveiling the intricacies of its atomic structure

Graphene, celebrated for its remarkable atomic structure, stands as a key material in diverse fields such as nanoelectronics and materials science. This paper presents a thorough investigation into the metric dimensions of graphene, emphasizing its unique spatial properties and their broad implications. We analyze how these metric dimensions affect graphene’s electrical, thermal, and mechanical behavior in various applications. Our study delves into the physical mechanisms behind these properties, including atomic interactions, lattice dynamics, and external factors like temperature and pressure. The metric dimension of a graph G is the minimum cardinality of a subset [Formula: see text] such that every pair of distinct vertices [Formula: see text] is uniquely identified by their distance vectors relative to W and the resolving set, ensures for any [Formula: see text] with [Formula: see text], there exists a vertex [Formula: see text] such that [Formula: see text], where d denotes the distance function in G. In this work, we will determine the metric dimension and Resolving set. However, these findings pave the way for innovative applications in electronics, composites, and beyond.