On the Metric Index of Circulant Networks–An Algorithmic Approach

A vertex v of a graph G uniquely determines (resolves) a pair (v
₁
, v
₂
) of vertices of G if the distance between v and v
₁
is different from the distance between v and v
₂
. The metric index is a distance-based topological index of a graph G, which is the least number of vertices in G chosen in such a way that each vertex of G 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.