EXTREMAL HYPER ZAGREB INDEX FOR TRICYCLIC GRAPHS

Miliˇcevi´c et al., in 2004, introduced topological indices known as Reformulated Zagreb indices, where they modified Zagreb indices using the edge-degree instead of vertex degree. In this paper, we present a simple approach to find the upper and lower bounds of the second reformulated Zagreb index, EM2(G), by using six graph operations/transformations. We prove that these operations significantly alter the value of reformulated Zagreb index. We apply these transformations and identify those graphs with cyclomatic number at most 3, namely trees, unicyclic, bicyclic and tricyclic graphs, which attain the upper and lower bounds of second reformulated Zagreb index for graphs.

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On Hyper-Zagreb Index of Three Graph Operations

Journal of Computer and Mathematical Sciences ◽

10.29055/jcms/1009 ◽

2019 ◽

Vol 10 (2) ◽

pp. 301-309

Author(s):

A. Bharali ◽

Amitav Doley

Keyword(s):

Zagreb Index ◽

Graph Operations

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On reduced second Zagreb index

Journal of Combinatorial Optimization ◽

10.1007/s10878-019-00518-7 ◽

2020 ◽

Vol 39 (3) ◽

pp. 776-791 ◽

Cited By ~ 1

Author(s):

Lkhagva Buyantogtokh ◽

Batmend Horoldagva ◽

Kinkar Chandra Das

Keyword(s):

Zagreb Index ◽

Second Zagreb Index

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On multiplicative degree based topological indices for planar octahedron networks

Main Group Metal Chemistry ◽

10.1515/mgmc-2020-0026 ◽

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.

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Eccentricity based topological indices of face centered cubic lattice FCC(n)

Main Group Metal Chemistry ◽

10.1515/mgmc-2021-0005 ◽

2020 ◽

Vol 44 (1) ◽

pp. 32-38

Author(s):

Hani Shaker ◽

Muhammad Imran ◽

Wasim Sajjad

Keyword(s):

Wiener Index ◽

Face Centered Cubic ◽

Eccentric Connectivity Index ◽

Zagreb Index ◽

Cubic Lattice ◽

Chemical Graph ◽

Chemical Graph Theory ◽

Wide Range ◽

Unit Cells ◽

Face Centered

Abstract Chemical graph theory has become a prime gadget for mathematical chemistry due to its wide range of graph theoretical applications for solving molecular problems. A numerical quantity is named as topological index which explains the topological characteristics of a chemical graph. Recently face centered cubic lattice FCC(n) attracted large attention due to its prominent and distinguished properties. Mujahed and Nagy (2016, 2018) calculated the precise expression for Wiener index and hyper-Wiener index on rows of unit cells of FCC(n). In this paper, we present the ECI (eccentric-connectivity index), TCI (total-eccentricity index), CEI (connective eccentric index), and first eccentric Zagreb index of face centered cubic lattice.

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Maximum total irregularity index of some families of graph with maximum degree n − 1

Asian-European Journal of Mathematics ◽

10.1142/s1793557122500693 ◽

2021 ◽

pp. 2250069

Author(s):

Shamaila Yousaf ◽

Akhlaq Ahmad Bhatti

Keyword(s):

Maximum Degree ◽

Discrete Math ◽

Simple Approach ◽

Unicyclic Graphs ◽

Irregularity Index ◽

Degree Of A Vertex ◽

Tricyclic Graphs ◽

Bicyclic Graphs ◽

Appl Math

The total irregularity index of a graph [Formula: see text] is defined by Abdo et al. [H. Abdo, S. Brandt and D. Dimitrov, The total irregularity of a graph, Discrete Math. Theor. Comput. Sci. 16 (2014) 201–206] as [Formula: see text], where [Formula: see text] denotes the degree of a vertex [Formula: see text]. In 2014, You et al. [L. H. You, J. S. Yang and Z. F. You, The maximal total irregularity of unicyclic graphs, Ars Comb. 114 (2014) 153–160.] characterized the graph having maximum [Formula: see text] value among all elements of the class [Formula: see text] (Unicyclic graphs) and Zhou et al. [L. H. You, J. S. Yang, Y. X. Zhu and Z. F. You, The maximal total irregularity of bicyclic graphs, J. Appl. Math. 2014 (2014) 785084, http://dx.doi.org/10.1155/2014/785084 ] characterized the graph having maximum [Formula: see text] value among all elements of the class [Formula: see text] (Bicyclic graphs). In this paper, we characterize the aforementioned graphs with an alternative but comparatively simple approach. Also, we characterized the graphs having maximum [Formula: see text] value among the classes [Formula: see text] (Tricyclic graphs), [Formula: see text] (Tetracyclic graphs), [Formula: see text] (Pentacyclic graphs) and [Formula: see text] (Hexacyclic graphs).

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Topological characterization of dendrimer, benzenoid, and nanocone

Zeitschrift für Naturforschung C ◽

10.1515/znc-2018-0153 ◽

2018 ◽

Vol 74 (1-2) ◽

pp. 35-43

Author(s):

Wei Gao ◽

Muhammad Kamran Siddiqui ◽

Najma Abdul Rehman ◽

Mehwish Hussain Muhammad

Keyword(s):

Chemical Properties ◽

Chemical Compounds ◽

Topological Indices ◽

Zagreb Index ◽

Topological Characterization ◽

Complex Molecules ◽

Chemical Structures ◽

Physico Chemical ◽

Quantitative Structure Activity Relationships

Abstract Dendrimers are large and complex molecules with very well defined chemical structures. More importantly, dendrimers are highly branched organic macromolecules with successive layers or generations of branch units surrounding a central core. Topological indices are numbers associated with molecular graphs for the purpose of allowing quantitative structure-activity relationships. These topological indices correlate certain physico-chemical properties such as the boiling point, stability, strain energy, and others, of chemical compounds. In this article, we determine hyper-Zagreb index, first multiple Zagreb index, second multiple Zagreb index, and Zagreb polynomials for hetrofunctional dendrimers, triangular benzenoids, and nanocones.

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Multiplicative Zagreb indices of cacti

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830916500403 ◽

2016 ◽

Vol 08 (03) ◽

pp. 1650040 ◽

Cited By ~ 6

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.

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On the first Zagreb Index of Polygraphs

Electronic Notes in Discrete Mathematics ◽

10.1016/j.endm.2013.11.028 ◽

2014 ◽

Vol 45 ◽

pp. 147-151 ◽

Cited By ~ 1

Author(s):

Hossein Shabani ◽

Reza Kahkeshani

Keyword(s):

First Zagreb Index ◽

Zagreb Index ◽

The First Zagreb Index

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

Mathematics ◽

10.3390/math6110227 ◽

2018 ◽

Vol 6 (11) ◽

pp. 227 ◽

Cited By ~ 4

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 .

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