Edge-Version Atom-Bond Connectivity and Geometric Arithmetic Indices of Generalized Bridge Molecular Graphs

Xiujun Zhang,Muhammad Jamil,Shehnaz Akhter,Jia-Bao Liu,Mohammad Farahani,Xinling Wu

doi:10.3390/sym10120751

Abstract

Topological indices are graph invariants computed by the distance or degree of vertices of the molecular graph. In chemical graph theory, topological indices have been successfully used in describing the structures and predicting certain physicochemical properties of chemical compounds. In this paper, we propose a definition of generalized bridge molecular graphs that can model more kinds of long chain polymerization products than the bridge molecular graphs, and provide some results of the edge versions of atom-bond connectivity ( A B C e ) and geometric arithmetic ( G A e ) indices for some generalized bridge molecular graphs, which have regular, periodic and symmetrical structures. The results of this paper offer promising prospects in the applications for chemical and material engineering, especially in chemical industry research.

Highlights

Let G be an undirected simple graph without loops or multiple edges
(called bridges), the generalized bridge molecular graph GBG ( G (1), v(1), G (2), v(2), · · ·, G (d), v(d) ; P(1), P(2), · · ·, P(d−1) ) is the graph obtained by connecting the vertices v(i) and v(i+1) by a path P(i) for which two end vertices are identified with v(i) and v(i+1) for i = 1, 2, ..., d − 1 (See Figure 3)
To describe more kinds of long chain polymerization products than the bridge molecular graphs, we propose the generalized bridge molecular graph structures

Summary

Introduction

Let G be an undirected simple graph without loops or multiple edges. We denote by V ( G ) the vertex set of G and we denote by E( G ) the edge set of G. G L be a line graph of G, so each vertex of L( G ) corresponds an edge of G. Symmetry 2018, 10, 751 denote corresponding vertices in line graph L( G ). Topological indices are graph invariants, which are obtained by performing some numerical operations on the distance or degree of vertices of the molecular graph. We define generalized bridge molecular graphs that could cover more kinds of long chain polymerization products, and the edge-version atom-bond connectivity and geometric arithmetic indices of generalized bridge molecular graphs are calculated. Are the degrees of the edges e1 and e2 in G

Definition of the Generalized Bridge Molecular Graph

Results and Discussion

Conclusions