scholarly journals The forgotten index of complement graph operations and its applications of molecular graph

2020 ◽  
Vol 3 (3) ◽  
pp. 53-61
Author(s):  
Mohammed Saad Alsharafi ◽  
◽  
Mahioub Mohammed Shubatah ◽  
Abdu Qaid Alameri ◽  
◽  
...  
Keyword(s):  
Molecular Graph ◽  
Cartesian Product ◽  
Numerical Parameter ◽  
Graph Operations ◽  
Strong Product ◽  
Complement Graph ◽  
Structure Activity ◽  
Forgotten Index ◽  
Quantitative Structure Activity Relationships ◽  
Mathematical Operations

A topological index of graph \(G\) is a numerical parameter related to graph which characterizes its molecular topology and is usually graph invariant. Topological indices are widely used to determine the correlation between the specific properties of molecules and the biological activity with their configuration in the study of quantitative structure-activity relationships (QSARs). In this paper some basic mathematical operations for the forgotten index of complement graph operations such as join \(\overline {G_1+G_2}\), tensor product \(\overline {G_1 \otimes G_2}\), Cartesian product \(\overline {G_1\times G_2}\), composition \(\overline {G_1\circ G_2}\), strong product \(\overline {G_1\ast G_2}\), disjunction \(\overline {G_1\vee G_2}\) and symmetric difference \(\overline {G_1\oplus G_2}\) will be explained. The results are applied to molecular graph of nanotorus and titania nanotubes.

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 The Second Hyper-Zagreb Index of Complement Graphs and Its Applications of Some Nano Structures

2021 ◽  
pp. 54-75
Author(s):  
Mohammed Alsharafi ◽  
Yusuf Zeren ◽  
Abdu Alameri
Keyword(s):  
Graph Theory ◽  
Cartesian Product ◽  
Quantitative Structure Activity Relationship ◽  
Quantitative Structure ◽  
Structure Property ◽  
Zagreb Index ◽  
Chemical Graph ◽  
Strong Product ◽  
Chemical Graph Theory ◽  
Mathematical Operations

In chemical graph theory, a topological descriptor is a numerical quantity that is based on the chemical structure of underlying chemical compound. Topological indices play an important role in chemical graph theory especially in the quantitative structure-property relationship (QSPR) and quantitative structure-activity relationship (QSAR). In this paper, we present explicit formulae for some basic mathematical operations for the second hyper-Zagreb index of complement graph containing the join G1 + G2, tensor product G1 \(\otimes\) G2, Cartesian product G1 x G2, composition G1 \(\circ\) G2, strong product G1 * G2, disjunction G1 V G2 and symmetric difference G1 \(\oplus\) G2. Moreover, we studied the second hyper-Zagreb index for some certain important physicochemical structures such as molecular complement graphs of V-Phenylenic Nanotube V PHX[q, p], V-Phenylenic Nanotorus V PHY [m, n] and Titania Nanotubes TiO2.

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

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Ahmed Ayache ◽  
Abdu Alameri ◽  
Mohammed Alsharafi ◽  
Hanan Ahmed
Keyword(s):  
Tensor Product ◽  
Topological Index ◽  
Molecular Graph ◽  
Cartesian Product ◽  
Explicit Formulas ◽  
Chemical Graph ◽  
Strong Product ◽  
Chemical Graphs ◽  
Product Composition ◽  
Graph Structures

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.

A novel graph invariant: The third leap Zagreb index under several graph operations

2019 ◽  
Vol 11 (05) ◽  
pp. 1950054 ◽  
Author(s):  
Durbar Maji ◽  
Ganesh Ghorai
Keyword(s):  
Tensor Product ◽  
Cartesian Product ◽  
Lexicographic Product ◽  
Graph Invariant ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
The Third ◽  
Graph Operations ◽  
Strong Product ◽  
Corona Product

The third leap Zagreb index of a graph [Formula: see text] is denoted as [Formula: see text] and is defined as [Formula: see text], where [Formula: see text] and [Formula: see text] are the 2-distance degree and the degree of the vertex [Formula: see text] in [Formula: see text], respectively. The first, second and third leap Zagreb indices were introduced by Naji et al. [A. M. Naji, N. D. Soner and I. Gutman, On leap Zagreb indices of graphs, Commun. Combin. Optim. 2(2) (2017) 99–117] in 2017. In this paper, the behavior of the third leap Zagreb index under several graph operations like the Cartesian product, Corona product, neighborhood Corona product, lexicographic product, strong product, tensor product, symmetric difference and disjunction of two graphs is studied.

scholarly journals The Sigma Coindex of Graph Operations

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Yasar Nacaroglu
Keyword(s):  
Tensor Product ◽  
Cartesian Product ◽  
Lexicographic Product ◽  
Nonadjacent Vertex ◽  
Graph Operations ◽  
Strong Product ◽  
Mathematical Properties ◽  
Corona Product

The sigma coindex is defined as the sum of the squares of the differences between the degrees of all nonadjacent vertex pairs. In this paper, we propose some mathematical properties of the sigma coindex. Later, we present precise results for the sigma coindices of various graph operations such as tensor product, Cartesian product, lexicographic product, disjunction, strong product, union, join, and corona product.

scholarly journals Sharp Bounds for the Inverse Sum Indeg Index of Graph Operations

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Anam Rani ◽  
Muhammad Imran ◽  
Usman Ali
Keyword(s):  
Tensor Product ◽  
Physicochemical Properties ◽  
Cartesian Product ◽  
Total Surface Area ◽  
Sharp Bounds ◽  
Graph Operations ◽  
Strong Product ◽  
Octane Isomers ◽  
International Academy ◽  
Corona Product

Vukičević and Gasperov introduced the concept of 148 discrete Adriatic indices in 2010. These indices showed good predictive properties against the testing sets of the International Academy of Mathematical Chemistry. Among these indices, twenty indices were taken as beneficial predictors of physicochemical properties. The inverse sum indeg index denoted by ISI G k of G k is a notable predictor of total surface area for octane isomers and is presented as ISI G k = ∑ g k g k ′ ∈ E G k d G k g k d G k g k ′ / d G k g k + d G k g k ′ , where d G k g k represents the degree of g k ∈ V G k . In this paper, we determine sharp bounds for ISI index of graph operations, including the Cartesian product, tensor product, strong product, composition, disjunction, symmetric difference, corona product, Indu–Bala product, union of graphs, double graph, and strong double graph.

scholarly journals The Reciprocal Complementary Wiener Number of Graph Operations

2021 ◽  
Vol 45 (01) ◽  
pp. 139-154
Author(s):  
R. NASIRI ◽  
A. NAKHAEI ◽  
A. R. SHOJAEIFARD
Keyword(s):  
Connected Graph ◽  
Cartesian Product ◽  
Symmetric Difference ◽  
Graph Operations ◽  
Strong Product ◽  
Product Composition ◽  
Wiener Number ◽  
Corona Product

The reciprocal complementary Wiener number of a connected graph G is defined as ∑ {x,y}⊆V (G) 1 D+1-−-dG(x,y), where D is the diameter of G and dG(x,y) is the distance between vertices x and y. In this work, we study the reciprocal complementary Wiener number of various graph operations such as join, Cartesian product, composition, strong product, disjunction, symmetric difference, corona product, splice and link of graphs.

scholarly journals Total Proper Connection and Graph Operations

2021 ◽  
pp. 2142006
Author(s):  
Yingying Zhang ◽  
Xiaoyu Zhu
Keyword(s):  
Connected Graph ◽  
Cartesian Product ◽  
Internal Vertex ◽  
Lexicographic Product ◽  
Colored Graph ◽  
Permutation Graphs ◽  
Connection Number ◽  
Graph Operations ◽  
Strong Product ◽  
Proper Connection Number

A graph is said to be total-colored if all the edges and vertices of the graph are colored. A path in a total-colored graph is a total proper path if (i) any two adjacent edges on the path differ in color, (ii) any two internal adjacent vertices on the path differ in color, and (iii) any internal vertex of the path differs in color from its incident edges on the path. A total-colored graph is called total-proper connected if any two vertices of the graph are connected by a total proper path of the graph. For a connected graph [Formula: see text], the total proper connection number of [Formula: see text], denoted by [Formula: see text], is defined as the smallest number of colors required to make [Formula: see text] total-proper connected. In this paper, we study the total proper connection number for the graph operations. We find that 3 is the total proper connection number for the join, the lexicographic product and the strong product of nearly all graphs. Besides, we study three kinds of graphs with one factor to be traceable for the Cartesian product as well as the permutation graphs of the star and traceable graphs. The values of the total proper connection number for these graphs are all [Formula: see text].

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