Metric Dimension of Cn(1, 2, 3) For n ≡ 0 (mod 6)

The metric dimension 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 metric basis for g and written dim(g) = ∥b∥. we have solved an open problem which shows dimension of circulant graph, dim(cn(1, 2, 3)) = 4, n ≡ 0 (mod 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.