Edge Distance‐based Topological Indices of Strength‐weighted Graphs and their Application to Coronoid Systems, Carbon Nanocones and SiO₂ Nanostructures

Abstract

The edge-Wiener index is conceived in analogous to the traditional Wiener index and it is defined as the sum of distances between all pairs of edges of a graph G. In the recent years, it has received considerable attention for determining the variations of its computation. Motivated by the method of computation of the traditional Wiener index based on canonical metric representation, we present the techniques to compute the edge-Wiener and vertex-edge-Wiener indices of G by dissecting the original graph G into smaller strength-weighted quotient graphs with respect to Djoković-Winkler relation. These techniques have been applied to compute the exact analytic expressions for the edge-Wiener and vertex-edge-Wiener indices of coronoid systems, carbon nanocones and SiO 2 nanostructures. In addition, we have reduced these techniques to the subdivision of partial cubes and applied to the circumcoronene series of benzenoid systems.

Relativistic structural characterization of molybdenum and tungsten disulfide materials

“…The formulae of several topological indices ( TI ) of a strength‐weighted graph G sw that are analyzed in this study are tabulated in Table 1, where TI ( G sw ) = TI ( G ) if w v = 1, s v = 0, w e = 1, and s e = 1. The computational techniques for evaluating these indices continues to be an interesting topic of research [ 23–25,31 ] because it facilitates the topological characterization without actually calculating the numerical parameters of the graph. The cut method is a classical computational procedure [ 31 ] employed to investigate topological indices and is being continuously revamped based on the kind of graphs.…”

“…[ 25–27,31 ] A coarser partition Θ c * is a partition { E 1 , …, E p } in which each set E i is the union of one or more Θ * classes of G . The formulae for determining the topological indices using this technique are summarized in the following theorem.Theorem [ 23–25 ] Let G sw = ( G , ( w v , s v ), ( w e , s e )) be a strength‐weighted graph . Let ℰ( G ) = { E 1 , E 2 , .…, E p } be a Θ c * partition of E ( G ), and TI ∈ { W , W e , W ev , Sz v , Sz e , Sz ev , PI , S , Gut , Mo , Mo e , Mo t , w + Mo , w + Mo e , w + Mo t , w * Mo , w * Mo e , w * Mo t }.…”

“…[22] Topological indices such as Szeged index, Gutman index, and Mostar index and their variants are extremely useful in characterizing the topostructural and peripheral properties of materials. [23][24][25][26][27][28] The above-mentioned topological indices can be computed in an elegant way by locating appropriate edge cuts, provided the given chemical material belongs to the family of partial cubes. However, the development of such techniques for a 2D complex molecular structure has been quite challenging due to computational complexities in the implementation process.…”

Mostar indices of SiO2 nanostructures and melem chain nanostructures

“…For instance, magnetic and polarizability properties of these materials are associated with their topological descriptors. In the past few decades, several studies have been conducted on the better implication of topological descriptors to enhance their significance 20,29–33 . Hence, the computation and estimation of topological descriptors of molecular structures are progressive trends of research, which have special significance in nanotechnology and theoretical chemistry.…”

The Topological Aspects of Phthalocyanines and Porphyrins Dendrimers

“…For further results related to different version of Mostar index, the interested readers is refereed to [26], [27], [29], [30], [32]- [34]. The evaluation and analysis of topological indices of molecular structures are modern trends of research, which are of the significant importance in nanotechnology and theoretical chemistry.…”