Damghan University Press Analytical and Numerical Solutions for Nonlinear Equations 3060-785X 8 2 2025 01 20 Metric Dimension of $C_n(1, 2, 3)$ for $ n \equiv 0 \pmod{6}$ 212 217 460 10.22128/ansne.2025.898.1121 EN Mostafa Mohagheghi Nejhad Adib Mazandaran Institute of Higher Education, Sari, Iran. 0000-0001-8529-0673 Journal Article 2024 10 29 The <em>metric dimension</em> of a connected graph $G$ is the minimum number of vertices in a subset $B$ of $G$ such that all other vertices are uniquely determined by their distances to the vertices in $B$‎. ‎In this case‎, ‎$B$ is called a \textit{metric basis} for $G$ and written $dim(G)=\Vert B\Vert$‎. ‎We have solved an open problem which shows dimension of circulant graph‎, ‎$dim(C_n(1,2,3))=4‎, ‎n \equiv 0 \pmod{6}$‎. ‎To prove this result‎, ‎we employ a combination of combinatorial techniques‎, ‎including distance-based analysis and structural properties of circulant graphs‎, ‎to carefully analyze the relationship between the graphs structure and its metric dimension‎. ‎The solution not only answers a previously unresolved question in graph theory but also provides valuable insights into the metric dimensions of more general classes of graphs‎, ‎particularly in network theory‎, ‎where understanding the metric dimension is essential for applications in sensor networks‎, ‎graph-based data storage‎, ‎and network routing‎. ‎This work lays the groundwork for future research on the metric dimensions of other families of graphs and has potential applications in optimizing communication and sensor placement in large-scale networks. https://ansne.du.ac.ir/article_460_f3003618255d98c2e428aa9a80dfa0c8.pdf