The Edge Version of Metric Dimension for the Family of Circulant Graphs Cₙ(1, 2)

Graph theory is widely used to analyze the structure models in chemistry, biology, computer science, operations research and sociology. Molecular bonds, species movement between regions, development of computer algorithms, shortest spanning trees in weighted graphs, aircraft scheduling and exploration of diffusion mechanisms are some of these structure models. Let $G = (V_{G}, E_{G})$ be a connected graph, where $V_{G}$ and $E_{G}$ represent the set of vertices and the set of edges respectively. The idea of the edge version of metric dimension is based on the distance of edges in a graph. Let $R_{E_{G}}$ be the smallest set of edges in a connected graph $G$ that forms a basis such that for every pair of edges $e_{1},e_{2}~\in ~E_{G}$ , there exists an edge $e~\in ~R_{E_{G}}$ for which $d_{E_{G}}(e_{1}, e)~\neq ~d_{E_{G}}(e_{2}, e)$ holds. In this paper, we show that the family of circulant graphs $C_{n}(1,2)$ is the family of graphs with constant edge version of metric dimension.