The Vertex Version of Weighted Wiener Number for Bicyclic Molecular Structures.

Wei Gao,Weifan Wang

doi:10.1155/2015/418106

Wei Gao, Weifan Wang

Open Access

https://doi.org/10.1155/2015/418106

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Abstract

Graphs are used to model chemical compounds and drugs. In the graphs, each vertex represents an atom of molecule and edges between the corresponding vertices are used to represent covalent bounds between atoms. We call such a graph, which is derived from a chemical compound, a molecular graph. Evidence shows that the vertex-weighted Wiener number, which is defined over this molecular graph, is strongly correlated to both the melting point and boiling point of the compounds. In this paper, we report the extremal vertex-weighted Wiener number of bicyclic molecular graph in terms of molecular structural analysis and graph transformations. The promising prospects of the application for the chemical and pharmacy engineering are illustrated by theoretical results achieved in this paper.

Highlights

The past 35 years have witnessed the conduction of investigations of degree or distance based topological indices
Topological indices’, which function as numerical descriptors of the molecular structure obtained from the corresponding molecular graph, several applications have been found in theoretical chemistry, especially in QSPR/QSAR study
One terminology issue we should clarify here is that the term “molecular graph” represents a chemical structure in many articles, whereas, in our paper, we consider the extremal vertexweighted Wiener number of bicyclic molecular graphs, and the bicyclic molecular graph which reaches the minimum vertex-weighted Wiener number may not represent a real molecular structure in chemical sciences

Summary

Introduction

The past 35 years have witnessed the conduction of investigations of degree or distance based topological indices. One terminology issue we should clarify here is that the term “molecular graph” represents a chemical structure in many articles, whereas, in our paper, we consider the extremal vertexweighted Wiener number of bicyclic molecular graphs, and the bicyclic molecular graph which reaches the minimum vertex-weighted Wiener number may not represent a real molecular structure in chemical sciences. The (singly) vertex-weighted Wiener number of molecular graph G was introduced by Doslic [9] as WV d This index has been shown to be strongly correlated to both the melting point and boiling point of the compounds. Several advances have been made in Wiener index, Zagreb index, PI index, hyper-Wiener index, and sum connectivity index of molecular graphs, the study of vertexweighted Wiener number for special chemical structures has been largely limited. We study the minimum vertex-weighted Wiener number of bicyclic molecular graphs

Notations and Useful Lemmas

Main Results and Proofs

Conclusion