The Second Hyper-Zagreb Coindex of Chemical Graphs and Some Applications

Ahmed Ayache,Mohammed Alsharafi,Hanan Ahmed,Abdu Alameri,Syed Ahtsham Haq Bokhary

doi:10.1155/2021/3687533

Abstract

The second hyper-Zagreb coindex is an efficient topological index that enables us to describe a molecule from its molecular graph. In this current study, we shall evaluate the second hyper-Zagreb coindex of some chemical graphs. In this study, we compute the value of the second hyper-Zagreb coindex of some chemical graph structures such as sildenafil, aspirin, and nicotine. We also present explicit formulas of the second hyper-Zagreb coindex of any graph that results from some interesting graphical operations such as tensor product, Cartesian product, composition, and strong product, and apply them on a q-multiwalled nanotorus.

Highlights

A graph can be identified by a corresponding numerical value, a sequence of numbers, or a special polynomial or a matrix
Special attention is directed to chemical graphs which constitute a wonderful topic in graph theory because of the abundance of applications in chemistry or in medical science [1, 2]
All graphs in this study are finite and simple, let G be a finite simple graph on V(G) n, vertices, and E(G) m, edges, and the degree of a vertex v is the number of edges event to v, denoted by δG(v). e complement of G, denoted by G, is a simple graph on the same set of vertices V(G), in which two vertices u and v are adjacent by an edge uv, if and only if they are not adjacent in G

Summary

Introduction

A graph can be identified by a corresponding numerical value, a sequence of numbers, or a special polynomial or a matrix. In 2016, Wei et al [12] defined new version of Zagreb topological indices It is called the hyper-Zagreb index that is defined as above. En, the second hyper-Zagreb index of a graph G is defined as the sum of the weights (δG(u)δG(v)) and is equal to HM2(G) 􏽘 δG(u)δG(v)􏼁2. We define a new version of Zagreb topological indices, based on the hyper-Zagreb index that is defined as above. It is called the second hyper-Zagreb index of a graph G and defined as the sum of the weights (δG(u)δG(v)), such that uv ∉ E(G) and is equal to uv ∉ E (G). Is study focused on one of the important topological coindices named the second hyper-Zagreb coindex. (iv) HM2(G1 ∗ G2) HM2(G2) [n1 + 10m1 + 10m1 (G1) +8m2(G1) + 6F(G1) +4ReZG3(G1) +Y(G1)] +HM2(G2)[n2 + 10m2 + 10m1(G2) + 8m2(G2) + 6F(G2) + 4ReZG3(G2) + Y(G2)] + Y(G2)[m1 + 2M1(G1) + 4M2(G1) + F(G1) + 2ReZG3 (G1)]Y (G1)[m2 + 2M1(G2) + 4M2(G2) + F(G2) + 2ReZG3(G2)] + 4ReZG3(G2) [m1+ 2M1 (G1) +2M2 (G1) + 2F(G1)] + 4ReZG3(G1) [m2 + 2M1(G2) + 2M2(G2) + 2F(G2)] + F(G2)[3M1 (G1) +8M2(G1)] + 4M1(G1)M2(G2) + 4M1(G2) M2 (G1) + 2HM2(G1)HM2(G2)+ 5F(G1)F(G2) + 6ReZG3(G1)ReZG3(G2)

Main Results

Applications

Conclusion