scholarly journals Maximizing and Minimizing Multiplicative Zagreb Indices of Graphs Subject to Given Number of Cut Edges

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 227 ◽  
Author(s):  
Shaohui Wang ◽  
Chunxiang Wang ◽  
Lin Chen ◽  
Jia-Bao Liu ◽  
Zehui Shao
Keyword(s):  
Molecular Graph ◽  
Zagreb Index ◽  
Zagreb Indices

Given a (molecular) graph, the first multiplicative Zagreb index Π 1 is considered to be the product of squares of the degree of its vertices, while the second multiplicative Zagreb index Π 2 is expressed as the product of endvertex degree of each edge over all edges. We consider a set of graphs G n , k having n vertices and k cut edges, and explore the graphs subject to a number of cut edges. In addition, the maximum and minimum multiplicative Zagreb indices of graphs in G n , k are provided. We also provide these graphs with the largest and smallest Π 1 ( G ) and Π 2 ( G ) in G n , k .

scholarly journals On multiplicative degree based topological indices for planar octahedron networks

2020 ◽  
Vol 43 (1) ◽  
pp. 219-228
Author(s):  
Ghulam Dustigeer ◽  
Haidar Ali ◽  
Muhammad Imran Khan ◽  
Yu-Ming Chu
Keyword(s):  
Graph Theory ◽  
Information Science ◽  
Biological Activities ◽  
Molecular Graph ◽  
Triangular Prism ◽  
Structure Property ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Chemical Graph Theory ◽  
Analytical Formulas

AbstractChemical graph theory is a branch of graph theory in which a chemical compound is presented with a simple graph called a molecular graph. There are atomic bonds in the chemistry of the chemical atomic graph and edges. The graph is connected when there is at least one connection between its vertices. The number that describes the topology of the graph is called the topological index. Cheminformatics is a new subject which is a combination of chemistry, mathematics and information science. It studies quantitative structure-activity (QSAR) and structure-property (QSPR) relationships that are used to predict the biological activities and properties of chemical compounds. We evaluated the second multiplicative Zagreb index, first and second universal Zagreb indices, first and second hyper Zagreb indices, sum and product connectivity indices for the planar octahedron network, triangular prism network, hex planar octahedron network, and give these indices closed analytical formulas.

scholarly journals Hyper Zagreb indices of composite graphs

2016 ◽  
Vol 4 (2) ◽  
pp. 47 ◽  
Author(s):  
Sharmila Devi ◽  
V. Kaladevi
Keyword(s):  
Molecular Graph ◽  
Sum Of Squares ◽  
First Zagreb Index ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Second Zagreb Index ◽  
The First Zagreb Index ◽  
Thorn Graph

For a (molecular) graph, the first Zagreb index M1 is equal to the sum of squares of the degrees of vertices, and the second Zagreb index M2 is equal to the sum of the products of the degrees of pairs of adjacent vertices. Similarly, the hyper Zagreb index is defined as the sum of square of degree of vertices over all the edges.  In this paper, First we obtain the hyper Zagreb indices of some derived graphs and the generalized transformations graphs. Finally, the hyper Zagreb indices of double, extended double, thorn graph, subdivision vertex corona of graphs, Splice and link graphs are obtained.

On the First and Second Zagreb and First and Second Hyper-Zagreb Indices of Carbon Nanocones CNCk[n]

2016 ◽  
Vol 13 (10) ◽  
pp. 7475-7482 ◽  
Author(s):  
Wei Gao ◽  
Mohammad Reza Farahani ◽  
Muhammad Kamran Siddiqui ◽  
Muhammad Kamran Jamil
Keyword(s):  
Molecular Graph ◽  
Topological Indices ◽  
Zagreb Index ◽  
Carbon Nanocones ◽  
Zagreb Indices ◽  
Minimum Path ◽  
Multiple Edges

Let G be a simple molecular graph without directed and multiple edges and without loops, the vertex and edge-sets of which are represented by V(G) and E(G), respectively. Suppose G is a connected molecular graph and vertices u, v ∈ V(>G). The distance dG(u,v) (or d(u,v) for short) between vertices u and V of G is defined as the length of a minimum path between u and V. The first and second Zagreb indices of a graph G are defined as M1(G) = ΣE=uv∈E(G)(dV+dV) and M2(G) = ΣE=uv∈E(G)(dV×dv) where du and dv are the degree of the vertices u and V of G. Recently the Hyper-Zagreb index of a graph G is defined as HM(G) = ΣE=uv∈E(G)(dV+dV)2, by Shirdel et al. In this paper, we define a new version of Zagreb topological indices, on based the Hyper-Zagreb index that defined as the sum of the weights (dudV)2 and the Second Hyper-Zagreb index of G is equal to HM2(G) = ΣE=uv∈E(G)(dVdV)22. In continue, exact formulas for the first and second Zagreb and Hyper-Zagreb indices of Carbon Nanocones CNCk[n] are computed.

scholarly journals Sharp bounds on Zagreb indices of cacti with k pendant vertices

Filomat ◽  
2012 ◽  
Vol 26 (6) ◽  
pp. 1189-1200 ◽  
Author(s):  
Shuchao Li ◽  
Huangxu Yang ◽  
Qin Zhao
Keyword(s):  
Perfect Matching ◽  
Molecular Graph ◽  
Common Vertex ◽  
First Zagreb Index ◽  
Zagreb Index ◽  
Sharp Bounds ◽  
Zagreb Indices ◽  
Sum Of Products ◽  
Vertex Degrees ◽  
The First Zagreb Index

For a (molecular) graph, the first Zagreb index M1 is equal to the sum of squares of its vertex degrees, and the second Zagreb index M2 is equal to the sum of products of degrees of pairs of adjacent vertices. A connected graph G is a cactus if any two of its cycles have at most one common vertex. In this paper, we investigate the first and the second Zagreb indices of cacti with k pendant vertices. We determine sharp bounds for M1 -, M2 -values of n-vertex cacti with k pendant vertices. As a consequence, we determine the n-vertex cacti with maximal Zagreb indices and we also determine the cactus with a perfect matching having maximal Zagreb indices.

scholarly journals The First and Second Zagreb Index of Complement Graph and Its Applications of Molecular Graph

2020 ◽  
pp. 15-30
Author(s):  
Mohammed S. Alsharafi ◽  
Mahioub M. Shubatah ◽  
Abdu Q. Alameri
Keyword(s):  
Tensor Product ◽  
Molecular Graph ◽  
Cartesian Product ◽  
Mathematical Operation ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Second Zagreb Index ◽  
Strong Product ◽  
Complement Graph ◽  
Graph Operation

In this paper, some basic mathematical operation for the second Zagreb indices of graph containing the join and strong product of graph operation, and the rst and second Zagreb indices of complement graph operations such as cartesian product G1 G2, composition G1 G2, disjunction G1 _ G2, symmetric dierence G1 G2, join G1 + G2, tensor product G1  G2, and strong product G1 G2 will be explained. The results are applied to molecular graph of nanotorus and titania nanotubes.

scholarly journals General Fifth M- Zagreb Polynomials of the TUC4C8(R)[p, q] 2D-Lattice and its Derived Graphs

2020 ◽  
Vol 10 (1) ◽  
pp. 1738-1747
Keyword(s):  
Chemical Compound ◽  
Molecular Graph ◽  
Vital Role ◽  
Topological Indices ◽  
Line Graphs ◽  
Topological Properties ◽  
Zagreb Index ◽  
Chemical Graph ◽  
Zagreb Indices ◽  
Physical Chemical

A molecular graph or a chemical graph is a graph related to the structure of a chemical compound. The topological indices play a vital role in understanding the physical, chemical, and topological properties of the respective compound. ln this article, we discuss the computation of the degree-based topological indices, namely - the fifth M-Zagreb indices and their polynomials, fifth hyper M-Zagreb indices and their polynomials, general fifth M-Zagreb indices and their polynomials, third Zagreb index and it is polynomial for the TUC_4 C_8 (R)[p,q] lattice, its subdivision, and para-line graphs.

On the Certain Topological Indices of Titania Nanotube TiO2[m, n]

2017 ◽  
Vol 72 (7) ◽  
pp. 647-654 ◽  
Author(s):  
M. Javaid ◽  
Jia-Bao Liu ◽  
M. A. Rehman ◽  
Shaohui Wang
Keyword(s):  
Chemical Compound ◽  
Topological Index ◽  
Molecular Graph ◽  
Topological Indices ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Mathematical Expressions ◽  
Titania Nanotube ◽  
Physical Features ◽  
Atom Bond

AbstractA numeric quantity that characterises the whole structure of a molecular graph is called the topological index that predicts the physical features, chemical reactivities, and boiling activities of the involved chemical compound in the molecular graph. In this article, we give new mathematical expressions for the multiple Zagreb indices, the generalised Zagreb index, the fourth version of atom-bond connectivity (ABC4) index, and the fifth version of geometric-arithmetic (GA5) index of TiO2[m, n]. In addition, we compute the latest developed topological index called by Sanskruti index. At the end, a comparison is also included to estimate the efficiency of the computed indices. Our results extended some known conclusions.

scholarly journals On the Multiplicative Degree-Based Topological Indices of Silicon-Carbon Si2C3-I[p,q] and Si2C3-II[p,q]

Symmetry ◽  
2018 ◽  
Vol 10 (8) ◽  
pp. 320 ◽  
Author(s):  
Young Kwun ◽  
Abaid Virk ◽  
Waqas Nazeer ◽  
M. Rehman ◽  
Shin Kang
Keyword(s):  
Molecular Structure ◽  
Graph Theory ◽  
Molecular Graph ◽  
Chemical Properties ◽  
Topological Indices ◽  
Molecular Compound ◽  
Zagreb Indices ◽  
Silicon Carbon ◽  
Physico Chemical ◽  
Structure Research

The application of graph theory in chemical and molecular structure research has far exceeded people’s expectations, and it has recently grown exponentially. In the molecular graph, atoms are represented by vertices and bonds by edges. Topological indices help us to predict many physico-chemical properties of the concerned molecular compound. In this article, we compute Generalized first and multiplicative Zagreb indices, the multiplicative version of the atomic bond connectivity index, and the Generalized multiplicative Geometric Arithmetic index for silicon-carbon Si2C3−I[p,q] and Si2C3−II[p,q] second.

scholarly journals Multiplicative Zagreb indices of cacti

2016 ◽  
Vol 08 (03) ◽  
pp. 1650040 ◽  
Author(s):  
Shaohui Wang ◽  
Bing Wei
Keyword(s):  
Graph Theory ◽  
Lower Bounds ◽  
Connected Graph ◽  
Upper And Lower Bounds ◽  
Extremal Graphs ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Material Chemistry ◽  
Cactus Graph ◽  
Cactus Graphs

Let [Formula: see text] be multiplicative Zagreb index of a graph [Formula: see text]. A connected graph is a cactus graph if and only if any two of its cycles have at most one vertex in common, which is a generalization of trees and has been the interest of researchers in the field of material chemistry and graph theory. In this paper, we use a new tool to obtain the upper and lower bounds of [Formula: see text] for all cactus graphs and characterize the corresponding extremal graphs.

scholarly journals Multiplicative Zagreb Indices of Molecular Graphs

2019 ◽  
Vol 2019 ◽  
pp. 1-19 ◽  
Author(s):  
Xiujun Zhang ◽  
H. M. Awais ◽  
M. Javaid ◽  
Muhammad Kamran Siddiqui
Keyword(s):  
Mathematical Modeling ◽  
Vital Role ◽  
Molecular Structures ◽  
Quantitative Structure ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Molecular Graphs ◽  
Structure Properties

Mathematical modeling with the help of numerical coding of graphs has been used in the different fields of science, especially in chemistry for the studies of the molecular structures. It also plays a vital role in the study of the quantitative structure activities relationship (QSAR) and quantitative structure properties relationship (QSPR) models. Todeshine et al. (2010) and Eliasi et al. (2012) defined two different versions of the 1st multiplicative Zagreb index as ∏Γ=∏p∈VΓdΓp2 and ∏1Γ=∏pq∈EΓdΓp+dΓq, respectively. In the same paper of Todeshine, they also defined the 2nd multiplicative Zagreb index as ∏2Γ=∏pq∈EΓdΓp×dΓq. Recently, Liu et al. [IEEE Access; 7(2019); 105479–-105488] defined the generalized subdivision-related operations of graphs and obtained the generalized F-sum graphs using these operations. They also computed the first and second Zagreb indices of the newly defined generalized F-sum graphs. In this paper, we extend this study and compute the upper bonds of the first multiplicative Zagreb and second multiplicative Zagreb indices of the generalized F-sum graphs. At the end, some particular results as applications of the obtained results for alkane are also included.

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