A Novel/Old Modification of the First Zagreb Index

AbstractFor a (molecular) graph G with vertex set V (G) and edge set E(G), the first Zagreb index of G is defined as, where dG(vi) is the degree of vertex vi in G. Recently Xu et al. introduced two graphical invariantsandnamed as first multiplicative Zagreb coindex and second multiplicative Zagreb coindex, respectively. The Narumi-Katayama index of a graph G, denoted by NK(G), is equal to the product of the degrees of the vertices of G, that is, NK(G) =. The irregularity index t(G) of G is defined as the number of distinct terms in the degree sequence of G. In this paper, we give some lower and upper bounds on the first Zagreb index M1(G) of graphs and trees in terms of number of vertices, irregularity index, maxi- mum degree, and characterize the extremal graphs. Moreover, we obtain some lower and upper bounds on the (first and second) multiplicative Zagreb coindices of graphs and characterize the extremal graphs. Finally, we present some relations between first Zagreb index and Narumi-Katayama index, and (first and second) multiplicative Zagreb index and coindices of graphs.

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Bounds on General Randić Index for F-Sum Graphs

Journal of Mathematics ◽

10.1155/2020/9129365 ◽

2020 ◽

Vol 2020 ◽

pp. 1-17

Author(s):

Xu Li ◽

Maqsood Ahmad ◽

Muhammad Javaid ◽

Muhammad Saeed ◽

Jia-Bao Liu

Keyword(s):

Molecular Graph ◽

Randić Index ◽

Lower And Upper Bounds ◽

First Zagreb Index ◽

Zagreb Index ◽

Randic Index ◽

Structure Activity ◽

General Randić Index ◽

General Randic Index ◽

The First Zagreb Index

A topological invariant is a numerical parameter associated with molecular graph and plays an imperative role in the study and analysis of quantitative structure activity/property relationships (QSAR/QSPR). The correlation between the entire π-electron energy and the structure of a molecular graph was explored and understood by the first Zagreb index. Recently, Liu et al. (2019) calculated the first general Zagreb index of the F-sum graphs. In the same paper, they also proposed the open problem to compute the general Randić index RαΓ=∑uv∈EΓdΓu×dΓvα of the F-sum graphs, where α∈R and dΓu denote the valency of the vertex u in the molecular graph Γ. Aim of this paper is to compute the lower and upper bounds of the general Randić index for the F-sum graphs when α∈N. We present numerous examples to support and check the reliability as well as validity of our bounds. Furthermore, the results acquired are the generalization of the results offered by Deng et al. (2016), who studied the general Randić index for exactly α=1.

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Nirmala energy

Open Journal of Discrete Applied Mathematics ◽

10.30538/psrp-odam2021.0055 ◽

2021 ◽

Vol 4 (2) ◽

pp. 11-16

Author(s):

Ivan Gutman ◽

◽

Veerabhadrappa R. Kulli ◽

Keyword(s):

Spectral Properties ◽

Topological Invariant ◽

Vertex Degree ◽

First Zagreb Index ◽

Zagreb Index ◽

Underlying Graph ◽

The First Zagreb Index

A novel vertex-degree-based topological invariant, called Nirmala index, was recently put forward, defined as the sum of the terms \(\sqrt{d(u)+d(v)}\) over all edges \(uv\) of the underlying graph, where \(d(u)\) is the degree of the vertex \(u\). Based on this index, we now introduce the respective ``Nirmala matrix'', and consider its spectrum and energy. An interesting finding is that some spectral properties of the Nirmala matrix, including its energy, are related to the first Zagreb index.

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Hyper Zagreb indices of composite graphs

International Journal of Advanced Mathematical Sciences ◽

10.14419/ijams.v6i1.8406 ◽

2016 ◽

Vol 4 (2) ◽

pp. 47 ◽

Cited By ~ 1

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.

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A Novel Weighted First Zagreb Index of Graph

Handbook of Research on Advanced Applications of Graph Theory in Modern Society - Advances in Computer and Electrical Engineering ◽

10.4018/978-1-5225-9380-5.ch004 ◽

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.

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More on the Laplacian Estrada index

Applicable Analysis and Discrete Mathematics ◽

10.2298/aadm0902371z ◽

2009 ◽

Vol 3 (2) ◽

pp. 371-378 ◽

Cited By ~ 17

Author(s):

Bo Zhou ◽

Ivan Gutman

Keyword(s):

Lower Bounds ◽

Upper And Lower Bounds ◽

First Zagreb Index ◽

Zagreb Index ◽

Laplacian Eigenvalues ◽

Estrada Index ◽

The First Zagreb Index

Let G be a graph with n vertices and let ?1, ?2, . . . , ?n be its Laplacian eigenvalues. In some recent works a quantity called Laplacian Estrada index was considered, defined as LEE(G)?n1 e?i. We now establish some further properties of LEE, mainly upper and lower bounds in terms of the number of vertices, number of edges, and the first Zagreb index.

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Sharp bounds for the first Zagreb index and first Zagreb coindex

Miskolc Mathematical Notes ◽

10.18514/mmn.2015.1274 ◽

2015 ◽

Vol 16 (2) ◽

pp. 1017-1024 ◽

Cited By ~ 5

Author(s):

Emina Milovanovic ◽

Igor Milovanovic

Keyword(s):

First Zagreb Index ◽

Zagreb Index ◽

Sharp Bounds ◽

The First Zagreb Index

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Topological Indices of Some New Graph Operations and Their Possible Applications

Asian Journal of Probability and Statistics ◽

10.9734/ajpas/2021/v13i230302 ◽

2021 ◽

pp. 16-30

Author(s):

Abdu Qaid Saif Alameri ◽

Mohammed Saad Yahya Al-Sharafi

Keyword(s):

Graph Theory ◽

Chemical Compounds ◽

Topological Indices ◽

First Zagreb Index ◽

Zagreb Index ◽

Structural Invariant ◽

Graph Operations ◽

Chemical Graph Theory ◽

Cyclic Alkanes ◽

The First Zagreb Index

A chemical graph theory is a fascinating branch of graph theory which has many applications related to chemistry. A topological index is a real number related to a graph, as its considered a structural invariant. It’s found that there is a strong correlation between the properties of chemical compounds and their topological indices. In this paper, we introduce some new graph operations for the first Zagreb index, second Zagreb index and forgotten index "F-index". Furthermore, it was found some possible applications on some new graph operations such as roperties of molecular graphs that resulted by alkanes or cyclic alkanes.

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Eccentricity-Based Topological Indices of a Cyclic Octahedron Structure

Mathematics ◽

10.3390/math6080141 ◽

2018 ◽

Vol 6 (8) ◽

pp. 141 ◽

Cited By ~ 3

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.

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