Give us a linked graph, $ G = (V, E). $ A vertex $ w\in V $ distinguishes between two components (vertices and edges) $ x, y\in E\cup V $ if $ d_G(w, x)\neq d_G (w, y). $ Let $ W_{1} $ and $ W_{2} $ be two resolving sets and $ W_{1} $ $ \neq $ $ W_{2} $. Then, we can say that the graph $ G $ has double resolving set. A nanotube derived from an quadrilateral-octagonal grid belongs to essential and extensively studied compounds in materials science. Nano-structures are very important due to their thickness. In this article, we have discussed the metric dimension of the graphs of nanotubes derived from the quadrilateral-octagonal grid. We proved that the generalized nanotube derived from quadrilateral-octagonal grid have three metric dimension. We also check that the exchange property is also held for this structure.

The exploration of edge metric dimension and its applications has been an ongoing discussion, particularly in the context of nanosheet graphs formed from the octagonal grid. Edge metric dimension is a concept that involves uniquely identifying the entire edge set of a structure with a selected subset from the vertex set, known as the edge resolving set. Let's consider two distinct edge resolving sets, denoted as $R_{e_1}$ and $R_{e_2}$, where $R_{e_1}\ne R_{e_2}$. In such instances, it indicates that the graph G possesses a double-edge resolving set. This implies the existence of two different subsets of the vertex set, each capable of uniquely identifying the entire edge set of the graph. In this article, we delve into the edge metric dimension of nanosheet graphs derived from the octagonal grid. Additionally, we initiate a discussion on the exchange property associated with the edge resolving set. The exchange property holds significance in the study of resolving sets, playing a crucial role in comprehending the structure and properties of the underlying graph.

The practical and theoretical significance of the resolvability parameter makes it an important factor, particularly in the context of network analysis. Its significance is seen in various applications and consequences: Network security, efficient routing, social network analysis, facility location, and site selection. This article finds the double resolvability parameters of the fosmidomycin anti-malaria drug. Resolvability parameters like double metric, double edge metric, and double mixed metric dimensions of fosmidomycin anti-malaria drug also hold exchange properties in the molecular graph of fosmidomycin. We convert the molecular structures of fosmidomycin into molecular graphs and then find some resolvability parameters.

The mixed metric dimension of a graph is an important parameter in characterizing its structural complexity, specifically in nanoscale networks where precision is paramount. In this article, we calculate the mixed metric dimension of hexagonal nano-network, a modal with significant applications in nanotechnology and material science. Through rigorous analysis, we set up that the mixed metric dimension of the hexagonal nano-network is exactly three, highlighting its minimal but enough resolving set that uniquely identifies all vertices. Furthermore, we check out the exchange property within this context, demonstrating the robust adaptability of the hexagonal network’s resolving sets. Our findings display that the alternate assets aren’t the handiest preserved but stronger in these nano-networks, allowing for flexible adjustments in resolving sets without compromising the network’s integrity. This examination offers critical insights into the fundamental properties of hexagonal nano-networks, offering a theoretical foundation for future research in nanomaterial design and optimization. The results underscore the potential of leveraging mixed metric dimensions and exchange properties to achieve efficient and scalable solutions in nano-network applications.

The exchange property in double-edge resolving partition sets is examined in this article, along with some real-world applications to city buildings. In graph theory, double-edge resolving sets are essential because they provide information on optimizing transportation and urban infrastructure. When utility units are switched out, the exchange property ensures the system is efficient and still works. City planners can create more adaptable and durable urban layouts by using this feature, guaranteeing that the best routes and shortest distances remain intact in various setups. We show how the exchange property in double-edge resolving partition sets can improve traffic management, emergency response systems, and overall urban planning through theoretical analysis and real-world case studies. The findings highlight the capability of graph-theoretical techniques in addressing complicated urban planning challenges, ultimately contributing to smarter, more sustainable town development. This study highlights the potential of advanced graph-theoretical concepts to address complex urban development challenges, contributing to the creation of smarter, more adaptive cities.