Analysis of Zagreb indices over zero-divisor graphs of commutative rings

2019 ◽  
Vol 12 (06) ◽  
pp. 2040003 ◽  
Author(s):  
Sümeyye Aykaç ◽  
Nihat Akgüneş ◽  
Ahmet Sinan Çevik
Keyword(s):  
Zero Divisor ◽  
Commutative Rings ◽  
First Zagreb Index ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Second Zagreb Index

In this paper, first Zagreb index, second Zagreb index, first multiplicative Zagreb index, second multiplicative Zagreb index, first Zagreb coindices index, second Zagreb coindices index, first multiplicative Zagreb coindices index, second multiplicative Zagreb coindices index of [Formula: see text] have been established, where [Formula: see text] and [Formula: see text] are prime.

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.

scholarly journals First and Second Zagreb Polynomials of VC5C7[p,q] and HC5C7[p,q]Nanotubes

2014 ◽  
Vol 31 ◽  
pp. 56-62 ◽  
Author(s):  
Mohammad Reza Farahani
Keyword(s):  
Graph Theory ◽  
Connected Graph ◽  
Topological Indices ◽  
First Zagreb Index ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Second Zagreb Index ◽  
Connectivity Indices ◽  
Anti Inflammatory ◽  
Simple Connected Graph

Let G = (V,E) be a simple connected graph. The sets of vertices and edges of G are denoted by V = V(G) and E = E(G), respectively. There exist many topological indices and connectivity indices in graph theory. The First and Second Zagreb indices were first introduced by Gutman and Trinajstić In1972. It is reported that these indices are useful in the study of anti-inflammatory activities of certain chemical instances, and in elsewhere. In this paper, we focus on the structure of ”G = VC5C7[p,q]”and ”H = HC5C7[p,q]” nanotubes and counting first Zagreb index Zg1(G) = ∑veVdv2 and Second Zagreb index Zg2(G) =∑e=uveE(G)(du·dv) of G and H, as well as First Zagreb polynomial Zg1(G,x ) =∑e=uveE(G)xdu+dv and Second Zagreb Polynomial Zg2(G,x) = ∑e=uveE(G)xdu·dv

scholarly journals Zagreb Connection Indices of Some Nanostructures

2018 ◽  
Vol 26 (2) ◽  
pp. 169-180 ◽  
Author(s):  
Saba Manzoor ◽  
Nisar Fatima ◽  
Akhlaq Ahmad Bhatti ◽  
Akbar Ali
Keyword(s):  
Electron Energy ◽  
Approximate Formula ◽  
Topological Indices ◽  
First Zagreb Index ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Second Zagreb Index ◽  
The First Zagreb Index ◽  
Molecular Branching

Abstract The first Zagreb index (occurred in an approximate formula of total π-electron energy, communicated in 1972) and the second Zagreb index (appeared in 1975, within the study of molecular branching) are among the most studied topological indices. Recently, three modified versions of the Zagreb indices were proposed independently in [A. Ali, N. Trinajstić, A novel/old modification of the first Zagreb index, arXiv:1705.10430 [math.CO], 2017] and [A. M. Naji, N. D. Soner, I. Gutman, On leap Zagreb indices of graphs, Commun. Comb. Optim., 2017, 2, 99–117], which were named as the Zagreb connection indices and the leap Zagreb indices, respectively. In this paper, we derive formulas for calculating these modified versions of the Zagreb indices of four well known nanostructures.

scholarly journals Topological indices of non-commuting graph of dihedral groups

2018 ◽  
Vol 14 ◽  
pp. 473-476 ◽  
Author(s):  
Nur Idayu Alimon ◽  
Nor Haniza Sarmin ◽  
Ahmad Erfanian
Keyword(s):  
Wiener Index ◽  
Topological Indices ◽  
First Zagreb Index ◽  
Dihedral Groups ◽  
Zagreb Index ◽  
Second Zagreb Index ◽  
Commuting Graph ◽  
Vertex Set ◽  
Group A ◽  
Group Of Symmetries

Assume  is a non-abelian group  A dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. The non-commuting graph of  denoted by  is the graph of vertex set  whose vertices are non-central elements, in which  is the center of  and two distinct vertices  and  are joined by an edge if and only if  In this paper, some topological indices of the non-commuting graph,  of the dihedral groups,  are presented. In order to determine the Edge-Wiener index, First Zagreb index and Second Zagreb index of the non-commuting graph,  of the dihedral groups,  previous results of some of the topological indices of non-commuting graph of finite group are used. Then, the non-commuting graphs of dihedral groups of different orders are found. Finally, the generalisation of Edge-Wiener index, First Zagreb index and Second Zagreb index of the non-commuting graphs of dihedral groups are determined.

scholarly journals On Leap Reduced Reciprocal Randic and Leap Reduced Second Zagreb Indices of Some Graphs

2019 ◽  
Vol 3 (2) ◽  
pp. 27-35
Author(s):  
Fazal Dayan ◽  
Muhammad Javaid ◽  
Muhammad Aziz ur Rehman
Keyword(s):  
Topological Indices ◽  
Randić Index ◽  
Zagreb Index ◽  
Randic Index ◽  
Zagreb Indices ◽  
Second Zagreb Index ◽  
Second Degree

Naji et al. introduced the leap Zagreb indices of a graph in 2017 which are new distance-degree-based topological indices conceived depending on the second degree of vertices. In this paper, we have defined the first and second leap reduced reciprocal Randic index and leap reduced second Zagreb index for selected wheel related graphs.

A Novel Weighted First Zagreb Index of Graph

2020 ◽  
pp. 92-103
Author(s):  
Jibonjyoti Buragohain ◽  
A. Bharali
Keyword(s):  
Connected Graph ◽  
Chemical Properties ◽  
Topological Indices ◽  
First Zagreb Index ◽  
Acentric Factor ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Physico Chemical ◽  
Octane Isomers ◽  
The First Zagreb Index

The Zagreb indices are the oldest among all degree-based topological indices. For a connected graph G, the first Zagreb index M1(G) is the sum of the term dG(u)+dG(v) corresponding to each edge uv in G, that is, M1 , where dG(u) is degree of the vertex u in G. In this chapter, the authors propose a weighted first Zagreb index and calculate its values for some standard graphs. Also, the authors study its correlations with various physico-chemical properties of octane isomers. It is found that this novel index has strong correlation with acentric factor and entropy of octane isomers as compared to other existing topological indices.

Degree-based topological indices of double graphs and strong double graphs

2017 ◽  
Vol 09 (05) ◽  
pp. 1750066 ◽  
Author(s):  
Muhammad Imran ◽  
Shehnaz Akhter
Keyword(s):  
Graph Theory ◽  
Physicochemical Properties ◽  
Boiling Point ◽  
Strain Energy ◽  
Chemical Compounds ◽  
Topological Indices ◽  
First Zagreb Index ◽  
Original Graph ◽  
Zagreb Index ◽  
Zagreb Indices

The topological indices are useful tools to the theoretical chemists that are provided by the graph theory. They correlate certain physicochemical properties such as boiling point, strain energy, stability, etc. of chemical compounds. For a graph [Formula: see text], the double graph [Formula: see text] is a graph obtained by taking two copies of graph [Formula: see text] and joining each vertex in one copy with the neighbors of corresponding vertex in another copy and strong double graph SD[Formula: see text] of the graph [Formula: see text] is the graph obtained by taking two copies of the graph [Formula: see text] and joining each vertex [Formula: see text] in one copy with the closed neighborhood of the corresponding vertex in another copy. In this paper, we compute the general sum-connectivity index, general Randi[Formula: see text] index, geometric–arithmetic index, general first Zagreb index, first and second multiplicative Zagreb indices for double graphs and strong double graphs and derive the exact expressions for these degree-base topological indices for double graphs and strong double graphs in terms of corresponding index of original graph [Formula: see text].

scholarly journals Two modified Zagreb indices for random structures

2021 ◽  
Vol 44 (1) ◽  
pp. 150-156
Author(s):  
Siman Li ◽  
Li Shi ◽  
Wei Gao
Keyword(s):  
Theoretical Basis ◽  
Chemical Properties ◽  
Random Structure ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Second Zagreb Index ◽  
Important Index ◽  
Physical Chemical ◽  
Random Structures

Abstract Random structure plays an important role in the composition of compounds, and topological index is an important index to measure indirectly the properties of compounds. The Zagreb indices and its revised versions (or redefined versions) are frequently used chemical topological indices, which provide the theoretical basis for the determination of various physical-chemical properties of compounds. This article uses the tricks of probability theory to determine the reduced second Zagreb index and hyper-Zagreb index of two kinds of vital random graphs: G(n, p) and G(n, m).

scholarly journals Eccentricity-Based Topological Indices of a Cyclic Octahedron Structure

Mathematics ◽  
2018 ◽  
Vol 6 (8) ◽  
pp. 141 ◽  
Author(s):  
Manzoor Zahid ◽  
Abdul Baig ◽  
Muhammad Naeem ◽  
Muhammad Azhar
Keyword(s):  
Connectivity Index ◽  
First Zagreb Index ◽  
Eccentric Connectivity Index ◽  
Zagreb Index ◽  
Chemical Graph ◽  
Second Zagreb Index ◽  
The Third ◽  
Average Eccentricity ◽  
The First Zagreb Index ◽  
Atom Bond

In this article, we study the chemical graph of a cyclic octahedron structure of dimension n and compute the eccentric connectivity polynomial, the eccentric connectivity index, the total eccentricity, the average eccentricity, the first Zagreb index, the second Zagreb index, the third Zagreb index, the atom bond connectivity index and the geometric arithmetic index of the cyclic octahedron structure. Furthermore, we give the analytically closed formulas of these indices which are helpful for studying the underlying topologies.

SOME RESULTS ON THE DIFFERENCE OF THE ZAGREB INDICES OF A GRAPH

2015 ◽  
Vol 92 (2) ◽  
pp. 177-186 ◽  
Author(s):  
MINGQIANG AN ◽  
LIMING XIONG
Keyword(s):  
Independence Number ◽  
Matching Number ◽  
Extremal Graphs ◽  
Vertex Connectivity ◽  
Zagreb Index ◽  
Sharp Bounds ◽  
Zagreb Indices ◽  
Second Zagreb Index ◽  
The Difference ◽  
Appl Math

The classical first and second Zagreb indices of a graph $G$ are defined as $M_{1}(G)=\sum _{v\in V(G)}d(v)^{2}$ and $M_{2}(G)=\sum _{e=uv\in E(G)}d(u)d(v),$ where $d(v)$ is the degree of the vertex $v$ of $G.$ Recently, Furtula et al. [‘On difference of Zagreb indices’, Discrete Appl. Math.178 (2014), 83–88] studied the difference of $M_{1}$ and $M_{2},$ and showed that this difference is closely related to the vertex-degree-based invariant $RM_{2}(G)=\sum _{e=uv\in E(G)}[d(u)-1][d(v)-1]$, the reduced second Zagreb index. In this paper, we present sharp bounds for the reduced second Zagreb index, given the matching number, independence number and vertex connectivity, and we also completely determine the extremal graphs.

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