Computation of Topological Indices of Some Graphs

Abstract

Let G = (V , E) be a simple connected graph with vertex set V and edge set E. The Wiener index of G is defined by Wwhere n eu (e|G) (resp. n ev (e|G)) is the number of edges of G closer to u (resp. v) than v (resp. u). In this paper we find the above indices for various graphs using the group of automorphisms of G. This is an efficient method of finding these indices especially when the automorphism group of G has a few orbits on V or E. We also find the Wiener indices of a few graphs which frequently arise in mathematical chemistry using inductive methods.

On the Edge Wiener index

“…One of the graphs that is important in graph theory, is hypercube Q r . In [7], by materials in group theory was showed that this graph is vertex-transitive and its Wiener index was computed. However now, acoording to concepts in graph theory, we can simply show that Q r is vertex-transitive.…”

“…However, because of the chemical facts about the Wiener index and also because it is an invariant of the graph, various researches found methods to calculate this index. Among the important works on finding the Wiener index of a general graph one is referred to the papers [7]- [11], [14]- [17], [24]- [26]. Another index that was recently defined in an analogous way, using the distance between the edges, is called the edge Wiener index [20].…”

The Hyper-Wiener Index of One-pentagonal Carbon Nanocone

“…But the definition of the Wiener index in terms of distances between vertices of a graph for the first time was given by Hosoya in [2].The Wiener index of graphs is extensively studied in [3], [4] and [5].In [6] new method are invented to calculate the Wiener index of a graph. a generalization of the concept of the Wiener index in recent chemical studies is the hyper-Wiener index.…”

Computation of Topological Indices of Some Special Graphs

“…Imran et al [7] studied the molecular topological properties and determined the analytical closed formula of the ABC, ABC 4 , ABC 5 , GA, GA 4 and GA 5 indices of Sierpinski networks. M. Darafsheh [18] developed different methods to calculate the Wiener index, Szeged index and Padmakar-Ivan index for various graphs using the group of automorphisms of G. He also found the Wiener indices of a few graphs using inductive methods. A. Ayache and A. Alameri [19] provided some topological indices of m k -graphs, such as the Wiener index, the hyper-Wiener index, the Wiener polarity, Zagreb indices, Schultz and modified Schultz indices and the Wiener-type invariant.…”