scholarly journals Computing Analysis of Connection-Based Indices and Coindices for Product of Molecular Networks

Symmetry ◽  
2020 ◽  
Vol 12 (8) ◽  
pp. 1320
Author(s):  
Usman Ali ◽  
Muhammad Javaid ◽  
Abdulaziz Mohammed Alanazi
Keyword(s):  
Carbon Nanotube ◽  
Electron Energy ◽  
Upper Bounds ◽  
Molecular Networks ◽  
Connection Number ◽  
Zagreb Indices ◽  
Linear Polynomial ◽  
Degree Of A Vertex ◽  
Octane Isomers ◽  
Alternant Hydrocarbons

Gutman and Trinajstić (1972) defined the connection-number based Zagreb indices, where connection number is degree of a vertex at distance two, in order to find the electron energy of alternant hydrocarbons. These indices remain symmetric for the isomorphic (molecular) networks. For the prediction of physicochemical and symmetrical properties of octane isomers, these indices are restudied in 2018. In this paper, first and second Zagreb connection coindices are defined and obtained in the form of upper bounds for the resultant networks in the terms of different indices of their factor networks, where resultant networks are obtained from two networks by the product-related operations, such as cartesian, corona, and lexicographic. For the molecular networks linear polynomial chain, carbon nanotube, alkane, cycloalkane, fence, and closed fence, first and second Zagreb connection coindices are computed in the consequence of the obtained results. An analysis of Zagreb connection indices and coindices on the aforesaid molecular networks is also included with the help of their numerical values and graphical presentations that shows the symmetric behaviour of these indices and coindices with in certain intervals of order and size of the under study (molecular) networks.

scholarly journals Computing Analysis for First Zagreb Connection Index and Coindex of Resultant Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-19 ◽  
Author(s):  
Muhammad Javaid ◽  
Usman Ali ◽  
Jia-Bao Liu
Keyword(s):  
Tensor Product ◽  
Physicochemical Properties ◽  
Electron Energy ◽  
Topological Index ◽  
Chemical Compounds ◽  
Upper Bounds ◽  
Symmetric Difference ◽  
Factor Graphs ◽  
Strong Product ◽  
Alternant Hydrocarbons

A numeric parameter which studies the behaviour, structural, toxicological, experimental, and physicochemical properties of chemical compounds under several graphs’ isomorphism is known as topological index. In 2018, Ali and Trinajstić studied the first Zagreb connection index Z C 1 to evaluate the value of a molecule. This concept was first studied by Gutman and Trinajstić in 1972 to find the solution of π -electron energy of alternant hydrocarbons. In this paper, the first Zagreb connection index and coindex are obtained in the form of exact formulae and upper bounds for the resultant graphs in terms of different indices of their factor graphs, where the resultant graphs are obtained by the product-related operations on graphs such as tensor product, strong product, symmetric difference, and disjunction. At the end, an analysis of the obtained results for the first Zagreb connection index and coindex on the aforesaid resultant graphs is interpreted with the help of numerical values and graphical depictions.

scholarly journals Zagreb Connection Indices of Molecular Graphs Based on Operations

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-15 ◽  
Author(s):  
Jinde Cao ◽  
Usman Ali ◽  
Muhammad Javaid ◽  
Chuangxia Huang
Keyword(s):  
Chemical Compounds ◽  
Total Electron ◽  
Connection Number ◽  
Absolute Value ◽  
Zagreb Indices ◽  
Molecular Graphs ◽  
Chemical Structures ◽  
Long Period ◽  
Linear Polynomial ◽  
Total Electron Energy

Topological index (numeric number) is a mathematical coding of the molecular graphs that predicts the physicochemical, biological, toxicological, and structural properties of the chemical compounds that are directly associated with the molecular graphs. The Zagreb connection indices are one of the TIs of the molecular graphs depending upon the connection number (degree of vertices at distance two) appeared in 1972 to compute the total electron energy of the alternant hydrocarbons. But after that, for a long period, these are not studied by researchers. Recently, Ali and Trinajstic Mol. Inform. 372018,1−7 restudied the Zagreb connection indices and reported that the Zagreb connection indices comparatively to the classical Zagreb indices provide the better absolute value of the correlation coefficient for the thirteen physicochemical properties of the octane isomers (all these tested values have been taken from the website http://www.moleculardescriptors.eu). In this paper, we compute the general results in the form of exact formulae & upper bounds of the second Zagreb connection index and modified first Zagreb connection index for the resultant graphs which are obtained by applying operations of corona, Cartesian, and lexicographic product. At the end, some applications of the obtained results for particular chemical structures such as alkanes, cycloalkanes, linear polynomial chain, carbon nanotubes, fence, and closed fence are presented. In addition, a comparison between exact and computed values of the aforesaid Zagreb indices is also included.

Mapping defects in a carbon nanotube by momentum transfer dependent electron energy loss spectromicroscopy

2012 ◽  
Vol 113 ◽  
pp. 158-164 ◽  
Author(s):  
Ebrahim Najafi ◽  
Adam P. Hitchcock ◽  
David Rossouw ◽  
Gianluigi A. Botton
Keyword(s):  
Carbon Nanotube ◽  
Energy Loss ◽  
Momentum Transfer ◽  
Electron Energy ◽  
Electron Energy Loss

Sharp upper bounds of F-index among bicyclic graphs

2021 ◽  
pp. 2250055
Author(s):  
R. Khoeilar ◽  
A. Jahanbani ◽  
L. Shahbazi ◽  
J. Rodríguez
Keyword(s):  
Maximum Degree ◽  
Topological Index ◽  
Upper Bounds ◽  
Degree Of A Vertex ◽  
Bicyclic Graphs

The [Formula: see text]-index of a graph [Formula: see text], denoted by [Formula: see text], is defined as the sum of weights [Formula: see text] over all edges [Formula: see text] of [Formula: see text], where [Formula: see text] denotes the degree of a vertex [Formula: see text]. In this paper, we give sharp upper bounds of the [Formula: see text]-index (forgotten topological index) over bicyclic graphs, in terms of the order and maximum degree.

scholarly journals Zagreb Connection Numbers for Cellular Neural Networks

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Jia-Bao Liu ◽  
Zahid Raza ◽  
Muhammad Javaid
Keyword(s):  
Neural Networks ◽  
Image Processing ◽  
Mathematical Models ◽  
Structural Properties ◽  
Cellular Neural Networks ◽  
Molecular Networks ◽  
Graph Invariants ◽  
Connection Number ◽  
Sensory Motor ◽  
3D Surfaces

Neural networks in which communication works only among the neighboring units are called cellular neural networks (CNNs). These are used in analyzing 3D surfaces, image processing, modeling biological vision, and reducing nonvisual problems of geometric maps and sensory-motor organs. Topological indices (TIs) are mathematical models of the (molecular) networks or structures which are presented in the form of numerical values, constitutional formulas, or numerical functions. These models predict the various chemical or structural properties of the under-study networks. We now consider analogous graph invariants, based on the second connection number of vertices, called Zagreb connection indices. The main objective of this paper is to compute these connection indices for the cellular neural networks (CNNs). In order to find their efficiency, a comparison among the obtained indices of CNN is also performed in the form of numerical tables and 3D plots.

scholarly journals Proper Vertex Connection and Graph Operations

2019 ◽  
Vol 19 (02) ◽  
pp. 1950001
Author(s):  
YINGYING ZHANG ◽  
XIAOYU ZHU
Keyword(s):  
Direct Product ◽  
Connected Graph ◽  
Upper Bounds ◽  
Lexicographic Product ◽  
Colored Graph ◽  
Connection Number ◽  
Graph Operations ◽  
Proper Vertex

A path in a vertex-colored graph is a vertex-proper path if any two internal adjacent vertices differ in color. A vertex-colored graph is proper vertex k-connected if any two vertices of the graph are connected by k disjoint vertex-proper paths of the graph. For a k-connected graph G, the proper vertex k-connection number of G, denoted by pvck(G), is defined as the smallest number of colors required to make G proper vertex k-connected. A vertex-colored graph is strong proper vertex-connected, if for any two vertices u, v of the graph, there exists a vertex-proper u-v geodesic. For a connected graph G, the strong proper vertex-connection number of G, denoted by spvc(G), is the smallest number of colors required to make G strong proper vertex-connected. In this paper, we study the proper vertex k-connection number and the strong proper vertex-connection number on the join of two graphs, the Cartesian, lexicographic, strong and direct product, and present exact values or upper bounds for these operations of graphs. Then we apply these results to some instances of Cartesian and lexicographic product networks.

scholarly journals Note on the Reformulated Zagreb Indices of Two Classes of Graphs

2020 ◽  
Vol 2020 ◽  
pp. 1-4
Author(s):  
Tongkun Qu ◽  
Mengya He ◽  
Shengjin Ji ◽  
Xia Li
Keyword(s):  
Upper Bounds ◽  
Extremal Graphs ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Vertex Degrees

The reformulated Zagreb indices of a graph are obtained from the original Zagreb indices by replacing vertex degrees with edge degrees, where the degree of an edge is taken as the sum of degrees of its two end vertices minus 2. In this paper, we obtain two upper bounds of the first reformulated Zagreb index among all graphs with p pendant vertices and all graphs having key vertices for which they will become trees after deleting their one key vertex. Moreover, the corresponding extremal graphs which attained these bounds are characterized.

scholarly journals Estimating and Approximating the Total π-Electron Energy of Benzenoid Hydrocarbons

2000 ◽  
Vol 55 (5) ◽  
pp. 507-512 ◽  
Author(s):  
I. Gutman ◽  
J. H. Koolen ◽  
V. Moulto ◽  
M. Parac ◽  
T. Soldatović ◽  
...  
Keyword(s):  
Electron Energy ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Benzenoid Hydrocarbons ◽  
Carbon Carbon Bonds ◽  
Carbon Bonds ◽  
Approximate Formulas ◽  
Better Than

Abstract Lower and upper bounds as well as approximate formulas for the total π-electron energy (E) of benzenoid hydrocarbons are deduced, depending only on the number of carbon atoms (n) and number of carbon-carbon bonds (to). These are better than the several previously known (n, m)-type estimates and approximations for E.

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