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 distancebased 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. INDEX TERMS Topological index, metric index, circulant networks.

Various iterative methods have been introduced by involving Taylor’s series on the auxiliary function g x to solve the nonlinear equation f x = 0 . In this paper, we introduce the expansion of g x with the inclusion of weights w i such that ∑ i = 1 p w i = 1 and knots τ i ∈ 0,1 in order to develop a new family of iterative methods. The methods proposed in the present paper are applicable for different choices of auxiliary function g x , and some already known methods can be viewed as the special cases of these methods. We consider the diverse scientific/engineering models to demonstrate the efficiency of the proposed methods.

The metric is a non-negative assignment to the pairs of nodes in a connected network N , which assigns the number of links lying in a smallest path between the nodes in the pairs. A pair (a, b) of nodes in N is said to be uniquely identified by a node c of N if the metric assigned to the pair (a, c) is different from the metric assigned to the pair (b, c). The metric index of N is the minimum number of nodes in N chosen in such a manner that every two nodes in N are uniquely identified by a chosen node. It is said to be constant for a family of networks if it remains unchange with the extension in the networks. In this paper, we consider a family of chordal ring networks and propose an algorithm which assistances in proving, with the aid of mathematical induction and the concept of good nodes, that there is no change in the metric index of chordal ring networks with the extension in the networks. INDEX TERMS Metric, metric index, chordal ring networks, good node.

A topological index is actually designed by transforming a chemical structure into a number. Topological index is a graph invariant which characterizes the topology of the graph and remains invariant under graph automorphism. Eccentricity based topological indices are of great importance and play a vital role in chemical graph theory. In this article, we consider a graph (non-zero component graph) associated to a finite dimensional vector space over a finite filed in the context of the following eleven eccentricity based topological indices: total eccentricity index; average eccentricity index; eccentric connectivity index; eccentric distance sum index; adjacent distance sum index; connective eccentricity index; geometric arithmetic index; atom bond connectivity index; and three versions of Zagreb indices. Relationship of the investigated indices and their dependency with respect to the involved parameters are also visualized by evaluating them numerically and by plotting their results.

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