Mixed metric dimension and exchange property of hexagonal nano-network

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.