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A vertex <inline-formula> <tex-math notation="LaTeX">$v$ </tex-math></inline-formula> of a graph <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> uniquely determines (resolves) a pair <inline-formula> <tex-math notation="LaTeX">$(v_{1},v_{2})$ </tex-math></inline-formula> of vertices of <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> if the distance between <inline-formula> <tex-math notation="LaTeX">$v$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$v_{1}$ </tex-math></inline-formula> is different from the distance between <inline-formula> <tex-math notation="LaTeX">$v$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$v_{2}$ </tex-math></inline-formula>. The metric index is a distance-based topological index of a graph <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula>, which is the least number of vertices in <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> chosen in such a way that each vertex of <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> can be determined uniquely by its distances to the chosen vertices. The metric index of a family of graphs is said to be constant if it does not increase with an increase in the number of vertices of graphs in the family. Otherwise, it is said to be unbounded. In this paper, we develop an algorithm to construct a larger order circulant network from the smallest order circulant network. Then, we consider two families of circulant networks: one in the context of constant metric index; and other in the perspective of the unbounded metric index to counter the popular belief that the metric index of circulant networks will never depend upon the number of vertices.

On the Metric Index of Circulant Networks–An Algorithmic Approach